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The "current-arc" data structure is a mechanism for visiting the in- and out-neighbors of a node in the flow network in a static circular order. If a singly linked list of neighbors is created for a node, the data structure can be as simple as a pointer into the list that steps through the list and
2895:
push–relabel algorithm organizes the active nodes into a queue. The initial active nodes can be inserted in arbitrary order. The algorithm always removes the node at the front of the queue for discharging. Whenever an inactive node becomes active, it is appended to the back of the queue.
2117:
must be equal to 0 at termination since there cannot be any remaining active nodes at the end of the algorithm's execution. This means that over the execution of the algorithm, the nonsaturating pushes must make up the difference of the relabel and saturating push operations in order for
169:. The algorithm was initially presented in November 1986 in STOC '86: Proceedings of the eighteenth annual ACM symposium on Theory of computing, and then officially in October 1988 as an article in the Journal of the ACM. Both papers detail a generic form of the algorithm terminating in
2691:
Based on the "current-arc" data structure, the discharge operation can be defined. A discharge operation applies on an active node and repeatedly pushes flow from the node until it becomes inactive, relabeling it as necessary to create admissible arcs in the process.
2922:
with respect to the admissible network. The algorithm scans the list from front to back and performs a discharge operation on the current node if it is active. If the node is relabeled, it is moved to the front of the list, and the scan is restarted from the front.
50:. The name "push–relabel" comes from the two basic operations used in the algorithm. Throughout its execution, the algorithm maintains a "preflow" and gradually converts it into a maximum flow by moving flow locally between neighboring nodes using
1884:
In order to bound the time complexity of the algorithm, we must analyze the number of push and relabel operations which occur within the main loop. The numbers of relabel, saturating push and nonsaturating push operations are analyzed separately.
2874:
Definition of the discharge operation reduces the push–relabel algorithm to repeatedly selecting an active node to discharge. Depending on the selection rule, the algorithm exhibits different time complexities. For the sake of brevity, we ignore
3046:
Heuristics are crucial to improving the empirical performance of the algorithm. Two commonly used heuristics are the gap heuristic and the global relabeling heuristic. The gap heuristic detects gaps in the labeling function. If there is a label
1481:
The generic push–relabel algorithm is used as a proof of concept only and does not contain implementation details on how to select an active node for the push and relabel operations. This generic version of the algorithm will terminate in
157:
and was published in 1974 in Soviet
Mathematical Dokladi 15. This pre-flow algorithm also used a push operation; however, it used distances in the auxiliary network to determine where to push the flow instead of a labeling system.
2401:, respectively, of the node during the execution of the algorithm. Each residual graph in the example only contains the residual arcs with a capacity larger than zero. Each residual graph may contain multiple iterations of the
1080:, and all other nodes are given a label of zero. Once the initialization is complete the algorithm repeatedly performs either the push or relabel operations against active nodes until no applicable operation can be performed.
213:
implementation using dynamic trees, and parallel/distributed implementation. As explained in, Goldberg-Tarjan introduced distance labels by incorporating them into the parallel maximum flow algorithm of Yossi
Shiloach and
145:. The idea of distance labels has led to a more efficient augmenting path algorithm, which in turn can be incorporated back into the push–relabel algorithm to create a variant with even higher empirical performance.
1589:. After initialisation, the algorithm repeatedly executes an applicable push or relabel operation until no such operations apply, at which point the pre-flow has been converted into a maximum flow.
2768:. The current-arc pointer only moves to the next neighbor when the edge to the current neighbor is saturated or non-admissible, and neither of these two properties can change until the active node
1823:. This can be proven by contradiction based on inequalities which arise in the labeling function when supposing that an augmenting path does exist. If the algorithm terminates, then all nodes in
3128:
to compute the exact labels of the nodes. Both heuristics skip unhelpful relabel operations, which are a bottleneck of the algorithm and contribute to the ineffectiveness of dynamic trees.
6882:
2945:
The highest-label push–relabel algorithm organizes all nodes into buckets indexed by their labels. The algorithm always selects an active node with the largest label to discharge.
3043:
time complexity regardless of the order of push and relabel operations and is therefore dominated by phase one. Alternatively, it can be implemented using flow decomposition.
2679:
or lower time complexity by enforcing appropriate rules in selecting applicable push and relabel operations. The empirical performance can be further improved by heuristics.
713:
551:
442:
393:
295:
6753:
6877:
2864:
2835:
2763:
2806:
2786:
1213:
since it uses up all the available capacity of the residual arc. Otherwise, all of the excess at the node is pushed across the residual arc. This is called an
1041:
The algorithm starts by creating a residual graph, initializing the preflow values to zero and performing a set of saturating push operations on residual arcs
6891:
3020:
The algorithm is typically separated into two phases. Phase one computes a maximum pre-flow by discharging only active nodes whose labels are below
1620:
is a valid labeling during its execution. This can be proven true by examining the effects of the push and relabel operations on the label function
6448:
Ahuja, Ravindra K.; Orlin, James B. (1991). "Distance-directed augmenting path algorithms for maximum flow and parametric maximum flow problems".
2340:
The following is a sample execution of the generic push-relabel algorithm, as defined above, on the following simple network flow graph diagram.
6746:
6685:
6642:
6405:
96:. Specific variants of the algorithms achieve even lower time complexities. The variant based on the highest label node selection rule has
6336:
Ahuja, Ravindra K.; Kodialam, Murali; Mishra, Ajay K.; Orlin, James B. (1997). "Computational investigations of maximum flow algorithms".
7452:
6914:
2643:
6966:
6827:
6582:
6309:
6263:
2788:
is relabelled. Therefore, when the pointer runs off, there are no admissible unsaturated edges and we have to relabel the active node
6934:
6471:
7045:
6739:
2892:
2918:
The relabel-to-front push–relabel algorithm organizes all nodes into a linked list and maintains the invariant that the list is
2360:
7322:
6984:
2633:
2608:
2575:
2555:
2535:
2511:
2482:
2441:
2425:
2109:
initially and stays nonnegative throughout the execution of the algorithm. Both relabels and saturating pushes can increase
1861:
obeys the flow conservation constraint and can be considered a normal flow. This flow is the maximum flow according to the
7473:
7430:
7050:
5473:
2626:
There are no remaining active nodes. The algorithm terminates and returns the maximum flow of the graph (as seen above).
7366:
7334:
59:
65:
The push–relabel algorithm is considered one of the most efficient maximum flow algorithms. The generic algorithm has a
7415:
7040:
6658:
Cherkassky, Boris V.; Goldberg, Andrew V. (1995). "On implementing push-relabel method for the maximum flow problem".
2348:
7361:
7317:
6919:
93:
7210:
6939:
6249:
1624:. The relabel operation increases the label value by the associated minimum plus one which will always satisfy the
687:, on nodes to determine which arcs should be selected for the push operation. This labeling function is denoted by
7100:
2494:
is relabeled in order to push its excess along its last remaining positive residual (i.e. push the excess back to
7478:
6762:
6701:
Derigs, U.; Meier, W. (1989). "Implementing
Goldberg's max-flow-algorithm ? A computational investigation".
31:
7285:
3137:
1862:
1068:. Similarly, the labels are initialized such that the label at the source is the number of nodes in the graph,
7147:
7329:
7228:
6944:
6822:
7420:
7405:
7295:
7173:
6799:
6766:
6663:
6457:
6383:
6345:
7309:
7275:
7178:
7120:
7001:
6807:
6787:
6625:
Cheriyan, J.; Maheshwari, S. N. (1988). "Analysis of preflow push algorithms for maximum network flow".
3024:. Phase two converts the maximum preflow into a maximum flow by returning excess flow that cannot reach
154:
3112:
immediately. The global relabeling heuristic periodically performs backward breadth-first search from
62:
performs global augmentations that send flow following paths from the source all the way to the sink.
7356:
7183:
7095:
6237:
3010:
2765:
2311:
nonsaturating pushes. Data structures can be designed to pick and execute an applicable operation in
6731:
6668:
6462:
6388:
6350:
1250:
to be the minimum value such that an admissible out-arc is created. Note that this always increases
7425:
7290:
7243:
7233:
7085:
7073:
6886:
6869:
6774:
6421:
Goldberg, Andrew V (1997). "An
Efficient Implementation of a Scaling Minimum-Cost Flow Algorithm".
6255:
2919:
66:
6378:
Goldberg, Andrew V. (2008). "The
Partial Augment–Relabel Algorithm for the Maximum Flow Problem".
696:
534:
425:
376:
278:
7160:
7129:
7115:
7105:
6896:
6718:
6552:
6502:
6315:
2967:
time complexity. If the lowest-label selection rule is used instead, the time complexity becomes
162:
115:
time complexity and is generally regarded as the benchmark for maximum flow algorithms. Subcubic
6812:
2421:
Initialise the residual graph by setting the preflow to values 0 and initialising the labeling.
1592:
generic-push-relabel(G, c, s, t): create a pre-flow f that saturates all out-arcs of s
7168:
6846:
6681:
6638:
6578:
6401:
6305:
6259:
142:
7248:
7238:
7142:
7019:
6924:
6906:
6859:
6770:
6710:
6673:
6630:
6607:
6542:
6494:
6467:
6430:
6393:
6355:
6297:
6233:
2083:
7264:
2840:
2811:
2739:
2587:
as the only remaining active node, but it cannot push its excess flow towards the sink.
7252:
7137:
7024:
6958:
6929:
6245:
2791:
2771:
716:. This function must satisfy the following conditions in order to be considered valid:
6359:
17:
7467:
7410:
7394:
6611:
166:
135:
6722:
6556:
6506:
6485:
Goldberg, Andrew V.; Tarjan, Robert E. (2014). "Efficient maximum flow algorithms".
6319:
6294:
Proceedings of the eighteenth annual ACM symposium on Theory of computing – STOC '86
1554:. At initialisation, the algorithm fulfills this requirement by creating a pre-flow
7348:
6854:
6598:
Shiloach, Yossi; Vishkin, Uzi (1982). "An O(n2log n) parallel max-flow algorithm".
658:
397:
232:
47:
43:
6397:
6292:
Goldberg, A V; Tarjan, R E (1986). "A new approach to the maximum flow problem".
7435:
6817:
2433:
Initial saturating push is performed across all preflow arcs out of the source,
215:
1924:
for all nodes. This means the relabel operation could potentially be performed
6241:
6677:
6634:
1435:, notice that this is trivially guaranteed by definition for the out-arcs of
1233:
which is neither the source nor the sink without any admissible out-arcs in
6472:
10.1002/1520-6750(199106)38:3<413::AID-NAV3220380310>3.0.CO;2-J
6434:
2226:, the total number of nonsaturating pushes must make up the difference of
6837:
2989:
Although in the description of the generic push–relabel algorithm above,
6547:
6530:
6301:
1876:
Therefore, the algorithm will return the maximum flow upon termination.
7157:
6714:
6382:. Lecture Notes in Computer Science. Vol. 5193. pp. 466–477.
2166:. Hence, the total contribution of all saturating pushes operations to
1764:
removes the corresponding constraint since the valid labeling property
54:
operations under the guidance of an admissible network maintained by
6498:
6627:
Foundations of
Software Technology and Theoretical Computer Science
2122:
to terminate with a value of 0. The relabel operation can increase
2101:
is the sum of the labels of all active nodes). It is obvious that
2082:
Bounding the number of nonsaturating pushes can be achieved via a
6662:. Lecture Notes in Computer Science. Vol. 920. p. 157.
1888:
In the algorithm, the relabel operation can be performed at most
1473:
can only satisfy the constraints less tightly, not violate them.
871:. As a result, if a valid labeling function exists, there are no
82:
time complexity, which is asymptotically more efficient than the
6629:. Lecture Notes in Computer Science. Vol. 338. p. 30.
2453:
is relabeled in order to push its excess flow towards the sink,
982: that are admissible. The admissible network is acyclic.
7392:
7208:
7071:
6999:
6785:
6735:
6983:
675:
The push–relabel algorithm uses a nonnegative integer valid
1916:
can never decrease, and the maximum label value is at most
3009:
at the beginning, it is preferable to perform a backward
2883:
when referring to the nodes in the following discussion.
2318:
time. Therefore, the time complexity of the algorithm is
2471:
in two subsequent saturating pushes; which still leaves
141:
The push–relabel algorithm has been extended to compute
6013:# send as much flow as possible to neighbours of source
2056:, and the total number of saturating pushes is at most
2009:
must first be relabeled, followed by a push on the arc
6001:# longest path from source to sink is less than n long
3161:#define MIN(X,Y) ((X) < (Y) ? (X) : (Y))
2843:
2814:
2794:
2774:
2742:
699:
537:
428:
379:
281:
6573:
Ahuja, R. K.; Magnanti, T. L.; Orlin, J. B. (1993).
2683:"Current-arc" data structure and discharge operation
2642:
The example (but with initial flow of 0) can be run
1857:
have no excess flow, and with no excess the preflow
1088:
The push operation applies on an admissible out-arc
153:
The concept of a preflow was originally designed by
7347:
7308:
7274:
7263:
7221:
7156:
7128:
7114:
7084:
7033:
7012:
6957:
6905:
6868:
6845:
6836:
6798:
6575:
2668:time complexity, efficient implementations achieve
2206:as in a saturating push. As a result, it decreases
6660:Integer Programming and Combinatorial Optimization
5743:# Find smallest new height making a push possible,
2858:
2829:
2800:
2780:
2757:
2714:relabel(u) rewind current-arc
1636:constraint. The push operation can send flow from
1329:remains a valid labeling function with respect to
707:
545:
436:
387:
289:
1604:there is an applicable push or relabel operation
863:is a lower bound of the unweighted distance from
816:is a lower bound of the unweighted distance from
2736:Finding the next admissible edge to push on has
2222:. Since relabels and saturating pushes increase
2033:increases by at least two. Therefore, there are
1313:> 0 𝓁 = 1 + min(𝓁 for all v such that c
1229:The relabel operation applies on an active node
892: because no such paths can be longer than
6529:Goldberg, Andrew V.; Tarjan, Robert E. (1988).
2504:is then removed from the set of active nodes.
2154:if it was inactive before the push, increasing
1258:and never creates a steep arc, which is an arc
5968:# we have checked all neighbours. must relabel
2688:rewinds to the head when it runs off the end.
1408:, where it effectively removes the constraint
6747:
2710:current-arc has run off the end of neighbors
2654:While the generic push–relabel algorithm has
2593:and then push its excess towards the source,
1803:exists then there is no augmenting path from
138:, although in practice it is less efficient.
8:
6531:"A new approach to the maximum-flow problem"
6524:
6522:
6520:
6518:
6516:
6373:
6371:
6369:
6287:
6285:
6283:
6281:
6279:
6277:
6275:
2733:current-arc point to the next neighbor of u
1309:> 0 and 𝓁 <= 𝓁 for all v such that c
1616:The algorithm maintains the condition that
161:The push-relabel algorithm was designed by
7389:
7305:
7271:
7218:
7205:
7125:
7081:
7068:
7009:
6996:
6842:
6795:
6782:
6754:
6740:
6732:
1969:Each saturating push on an admissible arc
1336:For a push operation on an admissible arc
1006:if it has positive excess with respect to
6667:
6546:
6461:
6387:
6349:
6331:
6329:
2842:
2813:
2793:
2773:
2741:
1728:will not affect the valid labeling since
701:
700:
698:
539:
538:
536:
430:
429:
427:
381:
380:
378:
283:
282:
280:
6988:Optimization computes maxima and minima.
6338:European Journal of Operational Research
6254:(2nd ed.). The MIT Press. pp.
6228:
6226:
6224:
5584:# residual capacity from u to v is C - F
2409:
1546:to satisfy the valid labeling condition
1423:To see that a relabel operation on node
6568:
6566:
6220:
2272:for the nonsaturating push operations.
1865:since there is no augmenting path from
800:In the algorithm, the label values of
134:time complexity can be achieved using
27:Algorithm in mathematical optimization
7184:Principal pivoting algorithm of Lemke
5614:# flow into node minus flow from node
3032:. It can be shown that phase two has
1519:, and there are no paths longer than
1164:> 0 and 𝓁 == 𝓁 + 1 Δ = min(x
7:
5746:# if such a push is possible at all.
5629:# neighbours seen since last relabel
2079:for the saturating push operations.
1993:. For the arc to be reinserted into
1904:times. This is because the labeling
1168:, c - f) f += Δ f -= Δ x
6703:Zeitschrift für Operations Research
2025:must be relabeled. In the process,
1325:After a push or relabel operation,
6828:Successive parabolic interpolation
2258:. This results in a time bound of
2068:. This results in a time bound of
1477:The generic push–relabel algorithm
449:function with respect to the flow
25:
7148:Projective algorithm of Karmarkar
2721:(u, v) = current-arc
1776:only applies to residual arcs in
7143:Ellipsoid algorithm of Khachiyan
7046:Sequential quadratic programming
6883:Broyden–Fletcher–Goldfarb–Shanno
2632:
2607:
2574:
2554:
2534:
2510:
2481:
2440:
2424:
2366:Final maximum flow network graph
2359:
2347:
2086:. We use the potential function
42:) is an algorithm for computing
6577:(1st ed.). Prentice Hall.
2914:Relabel-to-front selection rule
2616:Push the last bit of excess at
2275:In sum, the algorithm executes
1838:are not active. This means all
1600:𝓁 = 0 for all v ∈ V \ {s}
1558:that saturates all out-arcs of
967:is composed of the set of arcs
58:operations. In comparison, the
7101:Reduced gradient (Frank–Wolfe)
2853:
2847:
2824:
2818:
2808:, so having moved the pointer
2752:
2746:
1955:). This results in a bound of
1:
7431:Spiral optimization algorithm
7051:Successive linear programming
6360:10.1016/S0377-2217(96)00269-X
2997:is set to zero for each node
2005:for another saturating push,
1385:; it may also remove the arc
1179:A push operation that causes
194:sequential implementation, a
7169:Simplex algorithm of Dantzig
7041:Augmented Lagrangian methods
6612:10.1016/0196-6774(82)90013-X
6398:10.1007/978-3-540-87744-8_39
6248:(2001). "§26 Maximum flow".
2941:Highest label selection rule
2567:and then push its excess to
2547:and then push its excess to
2523:and then push its excess to
708:{\displaystyle \mathbb {N} }
546:{\displaystyle \mathbb {R} }
437:{\displaystyle \mathbb {R} }
388:{\displaystyle \mathbb {R} }
290:{\displaystyle \mathbb {R} }
3100:has been disconnected from
3061:for which there is no node
2870:Active node selection rules
2202:, but it can also activate
1966:for the relabel operation.
1570:is trivially valid for all
1321:Effects of push and relabel
847:has been disconnected from
7495:
6251:Introduction to Algorithms
6178:# start from front of list
5548:# C is the capacity matrix
2354:Initial flow network graph
2186:. A nonsaturating push on
1898: | − 2) < 2|
1550:must be disconnected from
1427:preserves the validity of
1053:exiting the source, where
559:residual capacity function
230:
7448:
7401:
7388:
7372:Push–relabel maximum flow
7217:
7204:
7174:Revised simplex algorithm
7080:
7067:
7008:
6995:
6981:
6794:
6781:
6487:Communications of the ACM
3164:#define INFINITE 10000000
3017:to compute exact labels.
2985:Implementation techniques
2837:times is paid for by the
2650:Practical implementations
2615:
2582:
2489:
2448:
667:with respect to the flow
561:with respect to the flow
227:Definitions and notations
32:mathematical optimization
6897:Symmetric rank-one (SR1)
6878:Berndt–Hall–Hall–Hausman
6678:10.1007/3-540-59408-6_49
6635:10.1007/3-540-50517-2_69
6450:Naval Research Logistics
5632:# node "queue"
5479:
3143:
3104:and can be relabeled to
2385:values denote the label
2236: | − 2) + (2|
2113:. However, the value of
1863:max-flow min-cut theorem
1305:relabel(u): assert x
1160:push(u, v): assert x
60:Ford–Fulkerson algorithm
7421:Parallel metaheuristics
7229:Approximation algorithm
6940:Powell's dog leg method
6892:Davidon–Fletcher–Powell
6788:Unconstrained nonlinear
6166:# move to front of list
2926:The algorithm also has
2729:push(u, v)
2138:. A saturating push on
7406:Evolutionary algorithm
6989:
6435:10.1006/jagm.1995.0805
5884:# check next neighbour
3132:Sample implementations
2860:
2831:
2802:
2782:
2759:
2413:Algorithm Operation(s)
2297:saturating pushes and
1811:in the residual graph
1608:execute the operation
709:
614:being the edges where
547:
438:
389:
358:vertices respectively,
291:
94:Edmonds–Karp algorithm
40:preflow–push algorithm
36:push–relabel algorithm
18:Push–relabel algorithm
7179:Criss-cross algorithm
7002:Constrained nonlinear
6987:
6808:Golden-section search
6600:Journal of Algorithms
6423:Journal of Algorithms
6380:Algorithms – ESA 2008
2861:
2832:
2803:
2783:
2760:
2725:(u, v) is admissible
2583:This leaves the node
2240: | − 1)(2|
2176: | − 1)(2|
2044:saturating pushes on
1981:removes the arc from
1795:and a valid labeling
1450:. For the in-arcs of
710:
548:
439:
390:
292:
155:Alexander V. Karzanov
7474:Network flow problem
7096:Cutting-plane method
3011:breadth-first search
2920:topologically sorted
2859:{\displaystyle O(V)}
2841:
2830:{\displaystyle O(V)}
2812:
2792:
2772:
2766:amortized complexity
2758:{\displaystyle O(1)}
2740:
2620:back to the source,
2373:In the example, the
2232: | − 1)(|
2132: | − 1)(|
1932:times for all nodes
1894: | − 1)(|
1348:, it may add an arc
697:
535:
426:
377:
279:
7426:Simulated annealing
7244:Integer programming
7234:Dynamic programming
7074:Convex optimization
6935:Levenberg–Marquardt
6548:10.1145/48014.61051
6302:10.1145/12130.12144
2887:FIFO selection rule
2866:relabel operation.
2248: |) ≤ 4|
2198:always deactivates
1912:value for any node
1740:. The deletion of
1219:non-saturating push
1149:units of flow from
679:which makes use of
67:strongly polynomial
7106:Subgradient method
6990:
6915:Conjugate gradient
6823:Nelder–Mead method
6715:10.1007/BF01415937
6535:Journal of the ACM
2948:The algorithm has
2899:The algorithm has
2856:
2827:
2798:
2778:
2755:
2695:discharge(u):
2475:with some excess.
2463:is then pushed to
2084:potential argument
1704:. The addition of
1100:of an active node
941:admissible network
839:is reachable from
705:
543:
434:
385:
287:
163:Andrew V. Goldberg
143:minimum cost flows
7461:
7460:
7444:
7443:
7384:
7383:
7380:
7379:
7343:
7342:
7304:
7303:
7200:
7199:
7196:
7195:
7192:
7191:
7063:
7062:
7059:
7058:
6979:
6978:
6975:
6974:
6953:
6952:
6687:978-3-540-59408-6
6644:978-3-540-50517-4
6407:978-3-540-87743-1
2937:time complexity.
2910:time complexity.
2801:{\displaystyle u}
2781:{\displaystyle u}
2640:
2639:
2404:perform operation
985:For a fixed flow
923: is called
783:Sink conservation
677:labeling function
261:capacity function
16:(Redirected from
7486:
7479:Graph algorithms
7390:
7306:
7272:
7249:Branch and bound
7239:Greedy algorithm
7219:
7206:
7126:
7082:
7069:
7010:
6997:
6945:Truncated Newton
6860:Wolfe conditions
6843:
6796:
6783:
6756:
6749:
6742:
6733:
6727:
6726:
6698:
6692:
6691:
6671:
6655:
6649:
6648:
6622:
6616:
6615:
6595:
6589:
6588:
6570:
6561:
6560:
6550:
6526:
6511:
6510:
6482:
6476:
6475:
6465:
6445:
6439:
6438:
6418:
6412:
6411:
6391:
6375:
6364:
6363:
6353:
6333:
6324:
6323:
6289:
6270:
6269:
6238:Leiserson, C. E.
6230:
6209:
6206:
6203:
6200:
6197:
6194:
6191:
6188:
6185:
6182:
6179:
6176:
6173:
6170:
6167:
6164:
6161:
6158:
6155:
6152:
6149:
6146:
6143:
6140:
6137:
6134:
6131:
6128:
6125:
6122:
6119:
6116:
6113:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6089:
6086:
6083:
6080:
6077:
6074:
6071:
6068:
6065:
6062:
6059:
6056:
6053:
6050:
6047:
6044:
6041:
6038:
6035:
6032:
6029:
6026:
6023:
6020:
6017:
6014:
6011:
6008:
6005:
6002:
5999:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5969:
5966:
5963:
5960:
5957:
5954:
5951:
5948:
5945:
5942:
5939:
5936:
5933:
5930:
5927:
5924:
5921:
5918:
5915:
5912:
5909:
5906:
5903:
5900:
5897:
5894:
5891:
5888:
5885:
5882:
5879:
5876:
5873:
5870:
5867:
5864:
5861:
5858:
5855:
5852:
5849:
5846:
5843:
5840:
5837:
5834:
5831:
5828:
5825:
5822:
5819:
5816:
5813:
5810:
5807:
5804:
5801:
5798:
5795:
5792:
5789:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5759:
5756:
5753:
5750:
5747:
5744:
5741:
5738:
5735:
5732:
5729:
5726:
5723:
5720:
5717:
5714:
5711:
5708:
5705:
5702:
5699:
5696:
5693:
5690:
5687:
5684:
5681:
5678:
5675:
5672:
5669:
5666:
5663:
5660:
5657:
5654:
5651:
5648:
5645:
5642:
5639:
5636:
5633:
5630:
5627:
5624:
5621:
5618:
5615:
5612:
5609:
5606:
5603:
5600:
5599:# height of node
5597:
5594:
5591:
5588:
5585:
5582:
5579:
5576:
5573:
5570:
5567:
5564:
5561:
5558:
5555:
5552:
5549:
5546:
5543:
5540:
5537:
5534:
5531:
5528:
5525:
5522:
5519:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5485:relabel_to_front
5483:
5466:
5463:
5460:
5457:
5454:
5451:
5448:
5445:
5442:
5439:
5436:
5433:
5430:
5427:
5424:
5421:
5418:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5292:
5289:
5286:
5283:
5280:
5277:
5274:
5271:
5268:
5265:
5262:
5259:
5256:
5253:
5250:
5247:
5244:
5241:
5238:
5235:
5232:
5229:
5226:
5223:
5220:
5217:
5214:
5211:
5208:
5205:
5202:
5199:
5196:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5160:
5157:
5154:
5151:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5109:
5106:
5103:
5100:
5097:
5094:
5091:
5088:
5085:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
5001:
4998:
4995:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4548:
4545:
4542:
4539:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3149:<stdlib.h>
3147:
3127:
3115:
3111:
3103:
3099:
3088:
3081:
3078:, then any node
3077:
3075:
3064:
3060:
3053:
3042:
3031:
3027:
3023:
3016:
3008:
3004:
2996:
2980:
2966:
2964:
2963:
2936:
2909:
2882:
2878:
2865:
2863:
2862:
2857:
2836:
2834:
2833:
2828:
2807:
2805:
2804:
2799:
2787:
2785:
2784:
2779:
2764:
2762:
2761:
2756:
2678:
2667:
2636:
2623:
2619:
2611:
2600:
2596:
2592:
2586:
2578:
2570:
2566:
2558:
2550:
2546:
2538:
2530:
2526:
2522:
2514:
2503:
2497:
2493:
2485:
2474:
2470:
2466:
2462:
2456:
2452:
2444:
2436:
2428:
2410:
2406:
2400:
2388:
2384:
2378:
2363:
2351:
2331:
2317:
2310:
2296:
2285:
2271:
2257:
2225:
2221:
2209:
2205:
2201:
2197:
2185:
2169:
2165:
2157:
2153:
2149:
2137:
2125:
2121:
2116:
2112:
2108:
2104:
2100:
2096:
2078:
2067:
2055:
2043:
2032:
2024:
2020:
2008:
2004:
1992:
1980:
1965:
1954:
1946:
1931:
1923:
1911:
1903:
1872:
1868:
1860:
1856:
1837:
1822:
1810:
1806:
1802:
1798:
1794:
1787:
1775:
1763:
1751:
1739:
1727:
1715:
1703:
1691:
1679:
1667:
1655:
1643:
1639:
1635:
1623:
1619:
1588:
1569:
1561:
1557:
1553:
1549:
1545:
1537:
1526:
1518:
1510:
1495:
1472:
1465:, the increased
1464:
1453:
1449:
1434:
1426:
1419:
1407:
1396:
1384:
1368:
1359:
1347:
1332:
1328:
1301:
1289:
1269:
1257:
1249:
1241:
1232:
1208:
1193:
1156:
1152:
1148:
1114:
1103:
1099:
1079:
1067:
1052:
1027:
1011:
1001:
990:
981:
966:
938:
922:
899:
891:
880:
870:
866:
862:
850:
846:
842:
838:
834:
823:
819:
815:
807:
803:
794:
777:
762:Source condition
756:
736:
715:
714:
712:
711:
706:
704:
670:
666:
660:residual network
656:
623:
613:
596:
564:
556:
552:
550:
549:
544:
542:
510:
452:
444:
443:
441:
440:
435:
433:
405:
395:
394:
392:
391:
386:
384:
349:
339:
325:
300:
296:
294:
293:
288:
286:
254:
212:
193:
182:
133:
114:
112:
111:
92:
81:
38:(alternatively,
21:
7494:
7493:
7489:
7488:
7487:
7485:
7484:
7483:
7464:
7463:
7462:
7457:
7440:
7397:
7376:
7339:
7300:
7277:
7266:
7259:
7213:
7188:
7152:
7119:
7110:
7087:
7076:
7055:
7029:
7025:Penalty methods
7020:Barrier methods
7004:
6991:
6971:
6967:Newton's method
6949:
6901:
6864:
6832:
6813:Powell's method
6790:
6777:
6760:
6730:
6700:
6699:
6695:
6688:
6669:10.1.1.150.3609
6657:
6656:
6652:
6645:
6624:
6623:
6619:
6597:
6596:
6592:
6585:
6572:
6571:
6564:
6528:
6527:
6514:
6499:10.1145/2628036
6484:
6483:
6479:
6463:10.1.1.297.5698
6447:
6446:
6442:
6420:
6419:
6415:
6408:
6389:10.1.1.150.5103
6377:
6376:
6367:
6351:10.1.1.297.2945
6335:
6334:
6327:
6312:
6296:. p. 136.
6291:
6290:
6273:
6266:
6232:
6231:
6222:
6218:
6213:
6212:
6211:
6210:
6207:
6204:
6201:
6198:
6195:
6192:
6189:
6186:
6183:
6180:
6177:
6174:
6171:
6168:
6165:
6162:
6159:
6156:
6153:
6150:
6147:
6144:
6141:
6138:
6135:
6132:
6129:
6126:
6123:
6120:
6117:
6114:
6111:
6108:
6105:
6102:
6099:
6096:
6093:
6090:
6087:
6084:
6081:
6078:
6075:
6072:
6069:
6066:
6063:
6060:
6057:
6054:
6051:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6027:
6024:
6021:
6018:
6015:
6012:
6009:
6006:
6003:
6000:
5997:
5994:
5991:
5988:
5985:
5982:
5979:
5976:
5973:
5970:
5967:
5964:
5961:
5958:
5955:
5952:
5949:
5946:
5943:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5919:
5916:
5913:
5910:
5907:
5904:
5901:
5898:
5895:
5892:
5889:
5886:
5883:
5880:
5877:
5874:
5871:
5868:
5865:
5862:
5859:
5856:
5853:
5850:
5847:
5844:
5841:
5838:
5835:
5832:
5829:
5826:
5823:
5820:
5817:
5814:
5811:
5808:
5805:
5802:
5799:
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5389:
5386:
5383:
5380:
5378:"Max Flow:
5377:
5374:
5371:
5368:
5365:
5362:
5359:
5356:
5353:
5350:
5348:"Capacity:
5347:
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5320:
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5245:
5243:// Sample graph
5242:
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3259:
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3250:
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3226:
3223:
3220:
3217:
3214:
3211:
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3205:
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3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3158:#define NODES 6
3157:
3155:<stdio.h>
3154:
3151:
3148:
3145:
3141:
3134:
3125:
3117:
3113:
3105:
3101:
3086:
3083:
3079:
3073:
3066:
3062:
3051:
3048:
3033:
3029:
3025:
3021:
3014:
3006:
3002:
2990:
2987:
2968:
2959:
2957:
2949:
2943:
2927:
2916:
2900:
2889:
2880:
2876:
2872:
2839:
2838:
2810:
2809:
2790:
2789:
2770:
2769:
2738:
2737:
2734:
2702:
2685:
2669:
2655:
2652:
2646:interactively.
2621:
2617:
2598:
2597:, via the node
2594:
2590:
2584:
2568:
2564:
2548:
2544:
2528:
2524:
2520:
2501:
2495:
2491:
2472:
2468:
2464:
2460:
2454:
2450:
2434:
2416:Residual Graph
2402:
2398:
2390:
2386:
2380:
2374:
2371:
2370:
2369:
2368:
2367:
2364:
2356:
2355:
2352:
2338:
2319:
2312:
2298:
2287:
2276:
2259:
2227:
2223:
2211:
2207:
2203:
2199:
2187:
2171:
2167:
2159:
2155:
2151:
2139:
2127:
2123:
2119:
2114:
2110:
2106:
2102:
2098:
2090:
2087:
2069:
2057:
2045:
2034:
2026:
2022:
2010:
2006:
2002:
1994:
1990:
1982:
1970:
1956:
1948:
1933:
1925:
1917:
1905:
1889:
1882:
1880:Time complexity
1870:
1866:
1858:
1839:
1824:
1820:
1812:
1808:
1804:
1800:
1796:
1792:
1785:
1777:
1765:
1761:
1753:
1741:
1729:
1725:
1717:
1705:
1701:
1693:
1681:
1680:and may delete
1677:
1669:
1657:
1656:. This may add
1645:
1641:
1637:
1625:
1621:
1617:
1614:
1609:
1571:
1563:
1559:
1555:
1551:
1547:
1539:
1538:, in order for
1536:
1528:
1520:
1512:
1500:
1483:
1479:
1466:
1463:
1455:
1451:
1448:
1440:
1428:
1424:
1409:
1406:
1398:
1386:
1370:
1366:
1361:
1349:
1337:
1330:
1326:
1323:
1318:
1316:
1312:
1308:
1291:
1279:
1271:
1259:
1251:
1243:
1239:
1234:
1230:
1227:
1211:saturating push
1195:
1180:
1177:
1175:
1171:
1167:
1163:
1154:
1150:
1138:
1125:
1116:
1113:
1105:
1101:
1089:
1086:
1069:
1054:
1042:
1039:
1034:
1021:
1013:
1007:
992:
986:
980:
968:
964:
951:
943:
928:
921:
904:
893:
890:
882:
872:
868:
864:
852:
848:
844:
840:
836:
833:
825:
821:
817:
809:
805:
801:
788:
767:
755:
738:
726:
695:
694:
688:
681:distance labels
668:
664:
654:
640:
632:
615:
608:
600:
574:
566:
562:
555:
533:
532:
522:
514:
497:
476:
462:
454:
450:
424:
423:
417:
409:
403:
375:
374:
361:
341:
331:
304:
299:
277:
276:
263:
241:
235:
229:
224:
195:
184:
170:
151:
116:
107:
105:
97:
83:
69:
28:
23:
22:
15:
12:
11:
5:
7492:
7490:
7482:
7481:
7476:
7466:
7465:
7459:
7458:
7456:
7455:
7449:
7446:
7445:
7442:
7441:
7439:
7438:
7433:
7428:
7423:
7418:
7413:
7408:
7402:
7399:
7398:
7395:Metaheuristics
7393:
7386:
7385:
7382:
7381:
7378:
7377:
7375:
7374:
7369:
7367:Ford–Fulkerson
7364:
7359:
7353:
7351:
7345:
7344:
7341:
7340:
7338:
7337:
7335:Floyd–Warshall
7332:
7327:
7326:
7325:
7314:
7312:
7302:
7301:
7299:
7298:
7293:
7288:
7282:
7280:
7269:
7261:
7260:
7258:
7257:
7256:
7255:
7241:
7236:
7231:
7225:
7223:
7215:
7214:
7209:
7202:
7201:
7198:
7197:
7194:
7193:
7190:
7189:
7187:
7186:
7181:
7176:
7171:
7165:
7163:
7154:
7153:
7151:
7150:
7145:
7140:
7138:Affine scaling
7134:
7132:
7130:Interior point
7123:
7112:
7111:
7109:
7108:
7103:
7098:
7092:
7090:
7078:
7077:
7072:
7065:
7064:
7061:
7060:
7057:
7056:
7054:
7053:
7048:
7043:
7037:
7035:
7034:Differentiable
7031:
7030:
7028:
7027:
7022:
7016:
7014:
7006:
7005:
7000:
6993:
6992:
6982:
6980:
6977:
6976:
6973:
6972:
6970:
6969:
6963:
6961:
6955:
6954:
6951:
6950:
6948:
6947:
6942:
6937:
6932:
6927:
6922:
6917:
6911:
6909:
6903:
6902:
6900:
6899:
6894:
6889:
6880:
6874:
6872:
6866:
6865:
6863:
6862:
6857:
6851:
6849:
6840:
6834:
6833:
6831:
6830:
6825:
6820:
6815:
6810:
6804:
6802:
6792:
6791:
6786:
6779:
6778:
6761:
6759:
6758:
6751:
6744:
6736:
6729:
6728:
6693:
6686:
6650:
6643:
6617:
6606:(2): 128–146.
6590:
6584:978-0136175490
6583:
6562:
6512:
6477:
6440:
6413:
6406:
6365:
6325:
6311:978-0897911931
6310:
6271:
6265:978-0262032933
6264:
6219:
6217:
6214:
5480:
5478:
5476:implementation
5472:
5471:
3144:
3142:
3140:implementation
3136:
3135:
3133:
3130:
3121:
3094:) < |
2986:
2983:
2942:
2939:
2915:
2912:
2888:
2885:
2871:
2868:
2855:
2852:
2849:
2846:
2826:
2823:
2820:
2817:
2797:
2777:
2754:
2751:
2748:
2745:
2700:
2694:
2684:
2681:
2651:
2648:
2638:
2637:
2629:
2628:
2613:
2612:
2604:
2603:
2580:
2579:
2572:
2560:
2559:
2552:
2540:
2539:
2532:
2516:
2515:
2507:
2506:
2487:
2486:
2478:
2477:
2459:The excess at
2446:
2445:
2438:
2430:
2429:
2422:
2418:
2417:
2414:
2394:
2365:
2358:
2357:
2353:
2346:
2345:
2344:
2343:
2342:
2337:
2334:
2252: ||
2244: ||
2180: ||
2089:
2062: ||
1998:
1986:
1881:
1878:
1816:
1781:
1757:
1721:
1697:
1673:
1613:
1610:
1591:
1562:, after which
1532:
1478:
1475:
1459:
1444:
1402:
1364:
1322:
1319:
1314:
1310:
1306:
1304:
1275:
1242:. It modifies
1237:
1226:
1223:
1173:
1169:
1165:
1161:
1159:
1134:
1121:
1109:
1085:
1082:
1038:
1037:Initialization
1035:
1033:
1030:
1017:
976:
960:
947:
917:
886:
829:
798:
797:
796:
795:
780:
779:
778:
759:
758:
757:
751:
721:Valid labeling
703:
673:
672:
649:
636:
626:
625:
604:
598:
570:
553:
541:
518:
512:
489:
468:
458:
432:
413:
407:
383:
359:
302:
297:
285:
231:Main article:
228:
225:
223:
220:
150:
147:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7491:
7480:
7477:
7475:
7472:
7471:
7469:
7454:
7451:
7450:
7447:
7437:
7434:
7432:
7429:
7427:
7424:
7422:
7419:
7417:
7414:
7412:
7411:Hill climbing
7409:
7407:
7404:
7403:
7400:
7396:
7391:
7387:
7373:
7370:
7368:
7365:
7363:
7360:
7358:
7355:
7354:
7352:
7350:
7349:Network flows
7346:
7336:
7333:
7331:
7328:
7324:
7321:
7320:
7319:
7316:
7315:
7313:
7311:
7310:Shortest path
7307:
7297:
7294:
7292:
7289:
7287:
7284:
7283:
7281:
7279:
7278:spanning tree
7273:
7270:
7268:
7262:
7254:
7250:
7247:
7246:
7245:
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7226:
7224:
7220:
7216:
7212:
7211:Combinatorial
7207:
7203:
7185:
7182:
7180:
7177:
7175:
7172:
7170:
7167:
7166:
7164:
7162:
7159:
7155:
7149:
7146:
7144:
7141:
7139:
7136:
7135:
7133:
7131:
7127:
7124:
7122:
7117:
7113:
7107:
7104:
7102:
7099:
7097:
7094:
7093:
7091:
7089:
7083:
7079:
7075:
7070:
7066:
7052:
7049:
7047:
7044:
7042:
7039:
7038:
7036:
7032:
7026:
7023:
7021:
7018:
7017:
7015:
7011:
7007:
7003:
6998:
6994:
6986:
6968:
6965:
6964:
6962:
6960:
6956:
6946:
6943:
6941:
6938:
6936:
6933:
6931:
6928:
6926:
6923:
6921:
6918:
6916:
6913:
6912:
6910:
6908:
6907:Other methods
6904:
6898:
6895:
6893:
6890:
6888:
6884:
6881:
6879:
6876:
6875:
6873:
6871:
6867:
6861:
6858:
6856:
6853:
6852:
6850:
6848:
6844:
6841:
6839:
6835:
6829:
6826:
6824:
6821:
6819:
6816:
6814:
6811:
6809:
6806:
6805:
6803:
6801:
6797:
6793:
6789:
6784:
6780:
6776:
6772:
6768:
6764:
6757:
6752:
6750:
6745:
6743:
6738:
6737:
6734:
6724:
6720:
6716:
6712:
6708:
6704:
6697:
6694:
6689:
6683:
6679:
6675:
6670:
6665:
6661:
6654:
6651:
6646:
6640:
6636:
6632:
6628:
6621:
6618:
6613:
6609:
6605:
6601:
6594:
6591:
6586:
6580:
6576:
6569:
6567:
6563:
6558:
6554:
6549:
6544:
6540:
6536:
6532:
6525:
6523:
6521:
6519:
6517:
6513:
6508:
6504:
6500:
6496:
6492:
6488:
6481:
6478:
6473:
6469:
6464:
6459:
6455:
6451:
6444:
6441:
6436:
6432:
6428:
6424:
6417:
6414:
6409:
6403:
6399:
6395:
6390:
6385:
6381:
6374:
6372:
6370:
6366:
6361:
6357:
6352:
6347:
6343:
6339:
6332:
6330:
6326:
6321:
6317:
6313:
6307:
6303:
6299:
6295:
6288:
6286:
6284:
6282:
6280:
6278:
6276:
6272:
6267:
6261:
6257:
6253:
6252:
6247:
6243:
6242:Rivest, R. L.
6239:
6235:
6234:Cormen, T. H.
6229:
6227:
6225:
6221:
6215:
5475:
3139:
3131:
3129:
3124:
3120:
3109:
3097:
3093:
3089:
3076:
3070:
3058:
3054:
3044:
3040:
3036:
3018:
3012:
3000:
2994:
2984:
2982:
2978:
2975:
2971:
2962:
2956:
2952:
2946:
2940:
2938:
2934:
2930:
2924:
2921:
2913:
2911:
2907:
2903:
2897:
2894:
2886:
2884:
2869:
2867:
2850:
2844:
2821:
2815:
2795:
2775:
2767:
2749:
2743:
2732:
2728:
2724:
2720:
2717:
2713:
2709:
2706:
2698:
2693:
2689:
2682:
2680:
2676:
2672:
2665:
2662:
2658:
2649:
2647:
2645:
2635:
2631:
2630:
2627:
2614:
2610:
2606:
2605:
2602:
2581:
2577:
2573:
2562:
2561:
2557:
2553:
2542:
2541:
2537:
2533:
2518:
2517:
2513:
2509:
2508:
2505:
2488:
2484:
2480:
2479:
2476:
2447:
2443:
2439:
2432:
2431:
2427:
2423:
2420:
2419:
2415:
2412:
2411:
2408:
2405:
2397:
2393:
2383:
2377:
2362:
2350:
2341:
2335:
2333:
2329:
2326:
2322:
2315:
2308:
2305:
2301:
2294:
2290:
2283:
2279:
2273:
2269:
2266:
2262:
2255:
2251:
2247:
2243:
2239:
2235:
2231:
2219:
2215:
2195:
2191:
2183:
2179:
2175:
2163:
2147:
2143:
2135:
2131:
2094:
2085:
2080:
2076:
2072:
2065:
2061:
2053:
2049:
2041:
2037:
2030:
2018:
2014:
2001:
1997:
1989:
1985:
1978:
1974:
1967:
1963:
1959:
1952:
1944:
1940:
1936:
1929:
1921:
1915:
1909:
1901:
1897:
1893:
1886:
1879:
1877:
1874:
1864:
1854:
1850:
1846:
1842:
1835:
1831:
1827:
1819:
1815:
1791:If a preflow
1789:
1784:
1780:
1773:
1769:
1760:
1756:
1749:
1745:
1737:
1733:
1724:
1720:
1713:
1709:
1700:
1696:
1689:
1685:
1676:
1672:
1665:
1661:
1653:
1649:
1633:
1629:
1611:
1607:
1603:
1599:
1596:𝓁 = |V|
1595:
1590:
1586:
1582:
1578:
1574:
1567:
1543:
1535:
1531:
1524:
1516:
1508:
1504:
1497:
1493:
1490:
1486:
1476:
1474:
1470:
1462:
1458:
1447:
1443:
1438:
1432:
1421:
1417:
1413:
1405:
1401:
1394:
1390:
1382:
1378:
1374:
1367:
1357:
1353:
1345:
1341:
1334:
1320:
1303:
1299:
1295:
1287:
1283:
1278:
1274:
1267:
1263:
1255:
1247:
1240:
1224:
1222:
1220:
1216:
1212:
1206:
1202:
1198:
1191:
1187:
1183:
1158:
1146:
1142:
1137:
1133:
1129:
1124:
1120:
1112:
1108:
1097:
1093:
1083:
1081:
1077:
1073:
1065:
1061:
1057:
1050:
1046:
1036:
1031:
1029:
1025:
1020:
1016:
1010:
1005:
999:
995:
989:
983:
979:
975:
971:
963:
959:
955:
950:
946:
942:
936:
932:
926:
920:
916:
912:
908:
901:
897:
889:
885:
879:
875:
860:
856:
832:
828:
813:
792:
787:
786:
784:
781:
775:
771:
766:
765:
763:
760:
754:
750:
746:
742:
734:
730:
725:
724:
722:
719:
718:
717:
692:
686:
682:
678:
662:
661:
652:
648:
644:
639:
635:
631:
630:
629:
622:
618:
612:
607:
603:
599:
594:
590:
586:
582:
578:
573:
569:
565:, defined by
560:
530:
526:
521:
517:
513:
508:
504:
500:
496:
492:
487:
483:
479:
475:
471:
466:
461:
457:
453:, defined by
448:
421:
416:
412:
408:
401:
400:
372:
368:
364:
360:
357:
353:
348:
344:
338:
334:
329:
323:
319:
315:
311:
307:
303:
274:
270:
266:
262:
258:
252:
248:
244:
240:
239:
238:
234:
226:
221:
219:
217:
210:
206:
202:
198:
191:
187:
183:along with a
180:
177:
173:
168:
167:Robert Tarjan
164:
159:
156:
148:
146:
144:
139:
137:
136:dynamic trees
131:
127:
123:
119:
110:
104:
100:
95:
90:
86:
79:
76:
72:
68:
63:
61:
57:
53:
49:
45:
44:maximum flows
41:
37:
33:
19:
7416:Local search
7371:
7362:Edmonds–Karp
7318:Bellman–Ford
7088:minimization
6920:Gauss–Newton
6870:Quasi–Newton
6855:Trust region
6763:Optimization
6706:
6702:
6696:
6659:
6653:
6626:
6620:
6603:
6599:
6593:
6574:
6538:
6534:
6490:
6486:
6480:
6453:
6449:
6443:
6426:
6422:
6416:
6379:
6341:
6337:
6293:
6250:
5432:"Flows:
3122:
3118:
3110: | + 1)
3107:
3095:
3091:
3085:
3072:
3068:
3056:
3055:< |
3050:
3045:
3038:
3034:
3019:
2998:
2992:
2988:
2976:
2973:
2969:
2960:
2954:
2950:
2947:
2944:
2932:
2928:
2925:
2917:
2905:
2901:
2898:
2890:
2873:
2735:
2730:
2726:
2722:
2718:
2715:
2711:
2707:
2704:
2696:
2690:
2686:
2674:
2670:
2663:
2660:
2656:
2653:
2641:
2625:
2588:
2499:
2490:Once again,
2458:
2403:
2395:
2391:
2381:
2375:
2372:
2339:
2327:
2324:
2320:
2313:
2306:
2303:
2299:
2292:
2288:
2281:
2277:
2274:
2267:
2264:
2260:
2253:
2249:
2245:
2241:
2237:
2233:
2229:
2217:
2213:
2210:by at least
2193:
2189:
2181:
2177:
2173:
2161:
2145:
2141:
2136: | − 2)
2133:
2129:
2092:
2081:
2074:
2070:
2063:
2059:
2051:
2047:
2039:
2035:
2028:
2016:
2012:
1999:
1995:
1987:
1983:
1976:
1972:
1968:
1961:
1957:
1950:
1942:
1938:
1934:
1930: | − 1
1927:
1919:
1913:
1907:
1899:
1895:
1891:
1887:
1883:
1875:
1852:
1848:
1844:
1840:
1833:
1829:
1825:
1817:
1813:
1790:
1782:
1778:
1771:
1767:
1758:
1754:
1747:
1743:
1735:
1731:
1722:
1718:
1711:
1707:
1698:
1694:
1687:
1683:
1674:
1670:
1663:
1659:
1651:
1647:
1631:
1627:
1615:
1605:
1601:
1597:
1593:
1584:
1580:
1576:
1572:
1565:
1541:
1533:
1529:
1522:
1514:
1506:
1502:
1498:
1491:
1488:
1484:
1480:
1468:
1460:
1456:
1445:
1441:
1436:
1430:
1422:
1415:
1411:
1403:
1399:
1392:
1388:
1380:
1376:
1372:
1362:
1355:
1351:
1343:
1339:
1335:
1324:
1297:
1293:
1285:
1281:
1276:
1272:
1265:
1261:
1253:
1245:
1235:
1228:
1218:
1215:unsaturating
1214:
1210:
1209:is called a
1204:
1200:
1196:
1189:
1185:
1181:
1178:
1144:
1140:
1135:
1131:
1127:
1122:
1118:
1110:
1106:
1095:
1091:
1087:
1075:
1071:
1063:
1059:
1055:
1048:
1044:
1040:
1023:
1018:
1014:
1008:
1003:
997:
993:
987:
984:
977:
973:
969:
961:
957:
953:
948:
944:
940:
934:
930:
924:
918:
914:
910:
906:
902:
895:
887:
883:
877:
873:
858:
854:
830:
826:
811:
799:
790:
782:
773:
769:
761:
752:
748:
744:
740:
732:
728:
720:
690:
684:
680:
676:
674:
659:
650:
646:
642:
637:
633:
627:
620:
616:
610:
605:
601:
592:
588:
584:
580:
576:
571:
567:
558:
528:
524:
519:
515:
506:
502:
498:
494:
490:
485:
481:
477:
473:
469:
464:
459:
455:
446:
419:
414:
410:
398:
370:
366:
362:
355:
351:
346:
342:
336:
332:
328:flow network
327:
321:
317:
313:
309:
305:
272:
268:
264:
260:
256:
250:
246:
242:
236:
233:Flow network
208:
204:
200:
196:
189:
185:
178:
175:
171:
160:
152:
140:
129:
125:
121:
117:
108:
102:
98:
88:
84:
77:
74:
70:
64:
55:
51:
48:flow network
39:
35:
29:
7436:Tabu search
6847:Convergence
6818:Line search
5444:printMatrix
5396:pushRelabel
5360:printMatrix
4805:printMatrix
4619:moveToFront
3989:pushRelabel
3860:moveToFront
3001:other than
2389:and excess
2170:is at most
2164: | − 1
2158:by at most
2126:by at most
1953: | − 2
1922: | − 1
1612:Correctness
1525: | − 1
1505:) = |
1379:) − 1 ≤ 𝓁(
1115:. It moves
1074:) = |
991:, a vertex
898: | − 1
857:) − |
808:are fixed.
772:) = |
657:denote the
557:denote the
445:denote the
350:are chosen
216:Uzi Vishkin
7468:Categories
7267:algorithms
6775:heuristics
6767:Algorithms
6709:(6): 383.
6541:(4): 921.
6456:(3): 413.
6344:(3): 509.
6216:References
6124:old_height
6094:old_height
5830:min_height
5812:min_height
5800:min_height
5749:min_height
5402:capacities
5366:capacities
5330:capacities
5318:capacities
5306:capacities
5294:capacities
5282:capacities
5270:capacities
5258:capacities
5246:capacities
5198:capacities
5069:capacities
5012:capacities
4610:old_height
4544:old_height
4340:&&
3716:&&
3524:min_height
3506:min_height
3494:min_height
3413:min_height
3065:such that
2286:relabels,
2150:activates
1296:) > 𝓁(
1270:such that
1172:-= Δ x
1032:Operations
1002:is called
925:admissible
835: if
689:𝓁 :
203: log(
7222:Paradigms
7121:quadratic
6838:Gradients
6800:Functions
6664:CiteSeerX
6493:(8): 82.
6458:CiteSeerX
6384:CiteSeerX
6346:CiteSeerX
6246:Stein, C.
6103:discharge
5842:discharge
4556:discharge
3548:discharge
3049:0 < 𝓁
2500:The node
2228:(2|
2172:(2|
2128:(2|
1890:(2|
1194:to reach
881:paths in
396:denote a
7453:Software
7330:Dijkstra
7161:exchange
6959:Hessians
6925:Gradient
6723:39730584
6557:52152408
6507:17014879
6429:: 1–29.
6320:14492800
6148:nodelist
6130:nodelist
6091:nodelist
6079:nodelist
5635:nodelist
4937:"%d
4406:INFINITE
3419:INFINITE
3152:#include
3146:#include
3106:(|
3090:< 𝓁(
2589:Relabel
2563:Relabel
2543:Relabel
2519:Relabel
2184: |)
2160:2|
2058:2|
1926:2|
1918:2|
1369:, where
1317:> 0)
1288:) > 0
1026:) > 0
1012:, i.e.,
737:for all
523: :
418: :
399:pre-flow
365: :
330:, where
222:Concepts
7296:Kruskal
7286:Borůvka
7276:Minimum
7013:General
6771:methods
5971:relabel
5731:relabel
4793:maxflow
4730:maxflow
4679:maxflow
3806:relabel
3329:relabel
3098: |
3059: |
2958:√
2703:> 0
2336:Example
2256: |
2216:) − 𝓁(
2066: |
2021:, then
1949:|
1947:(i.e.
1902: |
1770:) ≤ 𝓁(
1734:) = 𝓁(
1650:) = 𝓁(
1630:) ≤ 𝓁(
1521:|
1509: |
1414:) ≤ 𝓁(
1375:) = 𝓁(
1280: (
1225:Relabel
1184: (
1078: |
965: )
933:) = 𝓁(
903:An arc
894:|
861: |
851:, then
776: |
731:) ≤ 𝓁(
685:heights
591: (
575: (
257:network
149:History
106:√
56:relabel
7158:Basis-
7116:Linear
7086:Convex
6930:Mirror
6887:L-BFGS
6773:, and
6721:
6684:
6666:
6641:
6581:
6555:
6505:
6460:
6404:
6386:
6348:
6318:
6308:
6262:
6258:–698.
6196:return
6136:insert
6118:height
6100:height
6043:source
6004:excess
5992:height
5923:height
5917:height
5857:excess
5824:height
5818:height
5719:excess
5710:excess
5674:excess
5602:excess
5587:height
5497:source
5474:Python
5456:return
5438:"
5426:printf
5390:"
5372:printf
5354:"
5342:printf
5228:sizeof
5216:calloc
5186:sizeof
5174:calloc
5099:sizeof
5087:calloc
5054:sizeof
5042:calloc
4967:"
4961:"
4955:printf
4943:"
4931:printf
4790:return
4784:excess
4772:height
4604:height
4580:height
4574:excess
4550:height
4475:source
4469:excess
4400:excess
4388:height
4334:source
4259:sizeof
4244:calloc
4214:sizeof
4202:calloc
4172:sizeof
4160:calloc
4142:height
4130:sizeof
4118:calloc
4100:excess
4064:height
4055:excess
4031:source
3824:height
3755:excess
3728:height
3722:height
3641:excess
3605:height
3593:excess
3518:height
3512:height
3383:height
3311:excess
3299:excess
3257:excess
3215:excess
3126:
3071:) = 𝓁
2407:loop.
2399:
2097:(i.e.
2003:
1991:
1821:
1786:
1762:
1726:
1702:
1678:
1499:Since
1290:, and
1004:active
939:. The
653:
447:excess
352:source
34:, the
7357:Dinic
7265:Graph
6719:S2CID
6553:S2CID
6503:S2CID
6316:S2CID
6064:while
6025:range
5854:while
5767:range
5572:range
5521:->
5222:NODES
5180:NODES
5138:NODES
5093:NODES
5048:NODES
4916:NODES
4874:NODES
4820:const
4811:const
4715:NODES
4511:NODES
4499:while
4436:NODES
4394:NODES
4307:NODES
4250:NODES
4208:NODES
4166:NODES
4124:NODES
4004:const
3995:const
3668:NODES
3635:while
3563:const
3554:const
3449:NODES
3365:const
3356:const
3344:const
3335:const
3185:const
3176:const
3087:'
3082:with
3074:'
3052:'
3013:from
2697:while
2467:then
2449:Node
2220:) = 1
2088:Φ = Σ
1774:) + 1
1752:from
1738:) − 1
1692:from
1654:) + 1
1634:) + 1
1602:while
1568:) = 0
1517:) = 0
1418:) + 1
1397:from
1383:) + 1
1300:) + 1
1176:+= Δ
937:) + 1
843:. If
793:) = 0
735:) + 1
683:, or
619:<
488:) − Σ
467:) = Σ
259:with
255:be a
237:Let:
46:in a
7323:SPFA
7291:Prim
6885:and
6682:ISBN
6639:ISBN
6579:ISBN
6402:ISBN
6306:ISBN
6260:ISBN
6181:else
6121:>
6070:<
6037:push
5983:seen
5962:else
5953:seen
5947:else
5929:push
5920:>
5908:>
5893:seen
5875:<
5872:seen
5860:>
5791:>
5725:send
5716:send
5707:send
5698:send
5662:send
5644:push
5617:seen
5509:sink
5450:flow
5408:flow
5156:flow
5135:<
5024:flow
5003:flow
4988:void
4982:main
4913:<
4871:<
4802:void
4778:free
4766:free
4760:seen
4754:free
4748:list
4742:free
4712:<
4652:else
4631:list
4607:>
4586:seen
4535:list
4508:<
4451:push
4433:<
4361:list
4352:sink
4304:<
4226:list
4184:seen
4082:seen
4073:list
4040:sink
3977:temp
3935:>
3893:temp
3857:void
3836:seen
3800:else
3782:seen
3776:else
3737:push
3725:>
3707:>
3686:seen
3665:<
3662:seen
3644:>
3617:seen
3545:void
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3446:<
3326:void
3317:send
3305:send
3293:send
3281:send
3245:send
3170:push
3167:void
3005:and
2893:FIFO
2891:The
2879:and
2727:then
2716:else
2712:then
2644:here
2527:and
2379:and
1799:for
1117:min{
1084:Push
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356:sink
354:and
340:and
165:and
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52:push
7253:cut
7118:and
6711:doi
6674:doi
6631:doi
6608:doi
6543:doi
6495:doi
6468:doi
6431:doi
6394:doi
6356:doi
6298:doi
6256:643
6199:sum
6154:pop
6073:len
6016:for
5914:and
5839:def
5806:min
5758:for
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5668:min
5641:def
5563:for
5536:len
5524:int
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5503:int
5482:def
5423:));
5237:));
5234:int
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5114:for
5111:));
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5060:int
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3623:int
3611:int
3599:int
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3575:int
3557:int
3500:MIN
3425:for
3410:int
3401:int
3389:int
3377:int
3359:int
3338:int
3251:MIN
3242:int
3230:int
3221:int
3209:int
3197:int
3179:int
3116:in
3067:𝓁(
3028:to
2991:𝓁(
2731:let
2719:let
2498:).
2316:(1)
2212:𝓁(
2105:is
2091:𝓁(
2027:𝓁(
1937:\ {
1906:𝓁(
1869:to
1847:\ {
1828:\ {
1807:to
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1716:to
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1646:𝓁(
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1626:𝓁(
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1579:\ {
1564:𝓁(
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1360:to
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1252:𝓁(
1244:𝓁(
1217:or
1153:to
1130:),
1104:in
1070:𝓁(
1062:\ {
929:𝓁(
927:if
867:to
853:𝓁(
824:in
820:to
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789:𝓁(
768:𝓁(
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663:of
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308:= (
245:= (
30:In
7470::
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6765::
6717:.
6707:33
6705:.
6680:.
6672:.
6637:.
6602:.
6565:^
6551:.
6539:35
6537:.
6533:.
6515:^
6501:.
6491:57
6489:.
6466:.
6454:38
6452:.
6427:22
6425:.
6400:.
6392:.
6368:^
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6342:97
6340:.
6328:^
6314:.
6304:.
6274:^
6244:;
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6190:+=
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6115:if
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5956:+=
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5722:+=
5713:-=
5704:-=
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5581:)]
5569:in
5453:);
5441:);
5435:\n
5387:\n
5384:%d
5381:\n
5369:);
5357:);
5351:\n
5147:++
5081:**
5036:**
5009:**
5000:**
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4964:\n
4952:);
4940:\t
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4349:!=
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4325:((
4322:if
4319:){
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4253:-2
4247:((
4019:**
3947:--
3833:);
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3770:);
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3695:((
3692:if
3656:if
3578:**
3542:};
3515:);
3467:if
3458:++
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3302:-=
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3200:**
3084:𝓁
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2981:.
2723:if
2708:if
2705:do
2624:.
2601:.
2457:.
2387:𝓁
2332:.
2293:VE
2192:,
2144:,
2075:VE
2050:,
2015:,
1975:,
1941:,
1873:.
1851:,
1843:∈
1832:,
1797:𝓁
1788:.
1746:,
1710:,
1686:,
1662:,
1622:𝓁
1618:𝓁
1606:do
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1575:∈
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1496:.
1420:.
1391:,
1354:,
1342:,
1333:.
1327:𝓁
1302:.
1284:,
1264:,
1221:.
1203:,
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1147:)}
1094:,
1058:∈
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1028:.
1000:}
972:∈
956:,
945:G̃
909:,
900:.
785::
764::
743:,
723::
693:→
645:,
609:⊂
531:→
527:×
505:,
493:∈
484:,
472:∈
422:→
373:→
369:×
345:∈
335:∈
326:a
320:,
316:,
312:,
275:→
271:×
267::
249:,
218:.
211:))
201:VE
132:))
122:VE
89:VE
7251:/
6755:e
6748:t
6741:v
6725:.
6713::
6690:.
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6647:.
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6614:.
6610::
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6587:.
6559:.
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6322:.
6300::
6268:.
6208:)
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6193:1
6187:p
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6175:0
6172:=
6169:p
6160:p
6157:(
6151:.
6145:,
6142:0
6139:(
6133:.
6127::
6112:)
6109:u
6106:(
6097:=
6088:=
6085:u
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6067:p
6061:0
6058:=
6055:p
6052:)
6049:v
6046:,
6040:(
6031:n
6028:(
6019:v
6010:∞
6007:=
5998:n
5995:=
5989:0
5986:=
5980:)
5977:u
5974:(
5965::
5959:1
5950::
5944:)
5941:v
5938:,
5935:u
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5926::
5911:0
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5902:-
5899:C
5890:=
5887:v
5881::
5878:n
5866::
5863:0
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5845:(
5836:1
5833:+
5827:=
5821:)
5815:,
5809:(
5803:=
5797::
5794:0
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5785:-
5782:C
5773:n
5770:(
5761:v
5755:∞
5752:=
5737:u
5734:(
5701:F
5692:F
5689:)
5686:F
5683:-
5680:C
5677:,
5671:(
5665:=
5656:v
5653:,
5650:u
5647:(
5638:=
5626:n
5623:*
5620:=
5611:n
5608:*
5605:=
5596:n
5593:*
5590:=
5578:n
5575:(
5566:_
5560:n
5557:*
5554:=
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5542:C
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5533:=
5530:n
5527::
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5494:,
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5465:}
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5459:0
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5420:5
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5393:,
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5339:;
5336:4
5333:=
5327:;
5324:7
5321:=
5315:;
5312:7
5309:=
5303:;
5300:0
5297:=
5291:;
5288:0
5285:=
5279:;
5276:1
5273:=
5267:;
5264:9
5261:=
5255:;
5252:2
5249:=
5240:}
5231:(
5225:,
5219:(
5213:)
5210:*
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5201:=
5189:(
5183:,
5177:(
5171:)
5168:*
5162:(
5159:=
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5150:)
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5120:i
5117:(
5108:*
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5063:*
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4976:}
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4958:(
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4889:{
4886:)
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4853:(
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4682:=
4673:}
4670:}
4667:;
4664:1
4658:p
4655:{
4649:}
4646:;
4643:0
4640:=
4637:p
4628:,
4625:p
4622:(
4616:{
4613:)
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4592:u
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4553:;
4547:=
4538:;
4532:=
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4523:{
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4517:2
4514:-
4505:p
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4448:)
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4385:}
4382:}
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4364:=
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4337:)
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