Network%20Flows

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Network Flows

• Chun-Ta, Yu
• Graduate Institute
• Information Management Dept.
• National Taiwan University

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Chapter 12

• ASSIGHMENTS AND MATCHINGS

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Outline

• Introduction
• Bipartite Cardinality Matching Problem
• Bipartite Weighted Matching Problem
• Stable Marriage Problem
• Nonbipartite Cardinality Matching Problem

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Introduction

• Bipartite matching problem
• The cardinality problem
• The weighted problem
• Stable marriage problem
• Nonbipartite matching problem

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Bipartite Cardinality Matching Problem

• Transform this problem into a maximum flow
  problem in a simple network
• Each arc has a unit capacity and each node has an
  indegree of at most 1 or an outdegree of at most
  1
• Introduce a source node s and a sink node t
• Solving this problem at worst-case

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Bipartite Cardinality Matching Problem
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Bipartite Weighted Matching Problem

• Assignment problem
• Given a weighted bipartite network

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Bipartite Weighted Matching Problem

• The assignment problem can be viewed as
  adaptation of algorithm for the minimum cost flow
  problem
• Four algorithm to solve
• Successive Shortest Path Algorithm
• Hungarian Algorithm
• Relaxation Algorithm
• Cost Scaling Algorithm

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Successive Shortest Path Algorithm

• Discuss in section 9.7
• Augment 1 unit flow in every iteration
• Let S( n ,m ,C) denote the time needed to solve a
  shortest path problem with nonnegative arc
  lengths and n1N1
• The algorithm would terminate within n1
  iterations and would require O(n1S( n ,m ,C))

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Hungarian Algorithm

• Discuss in section 9.7, primal-dual algorithm
• With a single supply node s and a single demand
  node t.At every iteration ,the primal-dual
  algorithm computes shortest path distance from s
  to all other nodes
• The algorithm would terminate within n1
  iterations and would require O(n1S( n ,m ,C))

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Relaxation Algorithm

• Closely related to the successive shortest path
  algorithm
• Relax the constraint
• thus allowing any node in N2 to be assigned
  to more than one node in N1
• Overall running time O(n1S( n ,m ,C))

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Stable Marriage Problem

• A certain community consists of n men and n
  women. Each person ranks those of the opposite
  sex in accordance with his or her preferences for
  a spouse.
• Unstable if man-woman are not married to each
  other but prefer each other to their current
  spouses
• For any set of rankings, we can always find a
  stable matching in O(n2) time

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Stable Marriage Problem

• The input to the stable marriage problem consists
  of two n x n matrices.
• Each rank is an integer between 1 and n
• Priority list is a vector of n elements for each
  person, can be sorted using bucket sort in O(n2)
  time

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Stable marriage algorithm

• Propose-and-reject algorithm
• Time complexity
• Each iteration each woman receiving a proposal
  either (1) receives her first proposal (2)
  rejects some proposal
• Each woman rejects any mans proposal at most
  once, so total rejection times is (n-1) for each
  woman
• Total time is O(n2)
• Man-optimal matching

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Nonbipartite Cardinality Matching Problem

• Alternating Paths
• We refer to a path Pi1-i2-ik in the graph as an
  alternating path with respect to a matching M if
  every consecutive pair of arcs in the path
  contains one matched and one unmatched arc
• ex1-2-4-3-5

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Nonbipartite Cardinality Matching Problem

• Augmenting Paths
• We refer to an odd alternating path P with
  respect to matching M as an augmenting path if
  the first and last nodes in the path are
  unmatched
• ex1-2-4-3-5-6
• Interchanging the matched and unmatched arcs on
  augmenting paths can add one more cardinality

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Symmetric Difference

• Let S1 and S2 be two sets the symmetric
  difference of these sets, denoted S1?S2, is the
  set S1?S2 (S1?S2)-(S1nS2)
• for example, S1 4,5,7,8 and S2 2,4,8,9
  , then S1?S2 2,5,7,9

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Property 12.6

• If M is a matching and P is an augmenting path
  with respect to M, then M?P is a matching of
  cardinality M 1. Moreover, in the matching
  M?P, all the matched nodes in M remain matched
  and two additional nodes, namely the first and
  last nodes of P, are matched

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Property 12.7

• If M and M are two matchings, their symmetric
  difference defines the subgraph G (N, M?M)
  with the property that every component is one of
  the six types shown in Figure 12.7

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Augmenting Path Theorem

• If a node p is unmatched in a matching M, and
  this matching contains no augmenting path that
  starts at node p, then node p is unmatched in
  some maximum matching.

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Bipartite Matching Algorithm

• Start with a feasible matching M (which might be
  a null matching) and then repeat the following
  step for every unmatched node
• Try to identify an augmenting path starting at
  node p. If we find such a path P, replace M with
  M?P otherwise, delete node p and all the arcs
  incident to it from the graph

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Bipartite Matching Algorithm

• Search algorithm

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Bipartite Matching Algorithm
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Bipartite Matching Algorithm

• Time complexity
• The search algorithm execute at most n times
• For each node i, the search procedure performs
  one of the following two operations at most once
  (1) examine-even (2) examine-odd, the former
  operation require O(A(i)) time, about O(m)
• Total time O(nm)

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Unique label property

• A graph is said to possess a unique label
  property with respect to a given matching M and a
  root node p if the search procedure assigns a
  unique label to every labeled node irrespective
  of the order in which it examines labeled nodes

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• Bipartite network satisfy it nonbipartite
  network doesnt satisfy it

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Flower and Blossoms

• A flower, defined with respect to a matching M
  and a root node p, is a subgraph with two
  components
• Stem. A stem is an even length alternating path
  that starts at the root node p and terminates at
  some node w. We permit the possibility that p
  w, in which case we say that the stem is empty
• Blossom. A blossom is an odd length alternating
  cycle that starts and terminates at the terminal
  node w of a stem and has no other node in common
  with the stem. We refer to node w as the base of
  the blossom

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Flower and Blossoms
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Property 12.9

1. A stem spans 2l1 nodes and contains l matched
   arcs for some integer l?0
2. A blossom spans 2k1 nodes and contains k matched
   arcs for some integer k?1. The matched arcs match
   all nodes of the blossom except its base
3. The base of blossom is an even node

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Property 12.10

• Every node i in the blossom (except its base) is
  reachable from the root (or from the base of the
  blossom) through two distinct alternating paths.
  One has even length and the other has odd length

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Contracting a Blossom

• Introduce a new node b (pseudonode) and define
  its adjacency list A(b) A(i1)?A(i2)? ?A(jk)
• Update the adjacency list of every node
• by executing A(j)A(j)?b
• To be able to recover, set contracted node to
  inactive mode

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Nonbipartite matching algorithm
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Nonbipartite matching algorithm-- Find a
augmenting path
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Complexity of nonbipartite matching algorithm

• Lemma 12.13
• During an execution of the search procedure, the
  algorithm performs at most n/2 contractions
• Since each contraction adds at most one element
  to any adjacency list (the pseudonode), and since
  the algorithm performs at most n/2 contractions,
  no adjacency list will ever contain more than
  3n/2
• Each search procedure performs one of the
  following operation at most once (1)it discovers
  that node i is inactive (2)examine-odd(3)examine-e
  ven, (3) require Ac(i)?3n/2, so running time is
  O(n2)
• Total time complexity O(n3)

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Chapter 13

• MINIMUM SPANNING TREES

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Outline

• Introduction
• Optimality Condition
• Kruskals Algorithm
• Prims Algorithm
• Sollins Algorithm

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Introduction
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Optimality Condition

• Cut Optimality Conditions
• A Spanning tree T is a minimum spanning tree if
  and only if it satisfies the following cut
  optimality condition For every tree arc
  ,
• cij?ckl for every arc (k, l) contained in the
  cut formed by deleting arc (i, j) from T

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Optimality Condition

• Path Optimality Conditions
• A spanning tree T is a minimum spanning tree if
  and only if it satisfies the following path
  optimality conditions For every nontree arc (k,
  l) of G, cij?ckl for every arc (i, j) contained
  in the path in T connecting nodes k and l

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Kruskals Algorithm

• Time Complexity O(mn )

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Prims Algorithm

• Time Complexity O(m log n)

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Sollins Algorithm

• Time Complexity O(m log n)

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Summary of minimum spanning tree algorithm
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Matroids and the Greedy Algorithm

• Independenta subset I of objects do not form a
  cycle in the network
• Subset System (E, ?)a finite set of objects E
  and nonempty collection ? of subsets of these
  objects
• Matroida subset system satisfies the growth
  property that if Ip and Ip1 are independent sets
  containing p and p1 elements, we always can find
  an element , satisfying the
  property that Ip?e is an independent set

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Matroids and the Greedy Algorithm

• Maximal independent setan independent set I
  satisfying the property that we cannot add any
  other element e to I and produce another
  independent set
• Greedy algorithm

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Thank You!