Midterm

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Midterm
0 30-39 4 40-49
5 50-59 9 60-69 7 ?
70 3
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Matchings
matching M subgraph with
vertices of degree 1
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Matchings
unmatched vertex (vertex not in M)
matching M subgraph with
vertices of degree 1
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Matchings
unmatched vertex (vertex not in M)
matching M subgraph with
vertices of degree 1
perfect matching matching with no unmatched
vertices
(perfect world everybody has a buddy)
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Perfect matching
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Maximal / maximum
maximal no edge can be added to M to make a
bigger matching maximum there
is no matching with more edges
maximum ? maximal
maximal, NOT maximum
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Augmenting path
path connecting two unmatched vertices with
alternating non-matching and matching edges
If there exists augmenting path then the
matching is NOT maximum.
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Augmenting path
path connecting two unmatched vertices with
alternating non-matching and matching edges
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Augmenting path
path connecting two unmatched vertices with
alternating non-matching and matching edges
Lucky ?
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Augmenting path
Lucky ?
NO
Theorem If matching M is not maximum then
there exists an augmenting path.
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Augmenting path
Theorem If the matching M is not maximum then
there exists an augmenting path.
M matching, not maximum B bigger matching
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Proof
M matching, not maximum B bigger matching
M?B
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Proof
M matching, not maximum B bigger matching
M?B
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Proof
M matching, not maximum B bigger matching
M?B
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Proof
M matching, not maximum B bigger matching
M?B
vertices of degree 0,1,2 ? paths and cycles no
odd cycles must have a path with more edges from
B than from M ? augmenting path
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How to find maximum matching?
M?0 repeat find augmenting path P M ?
M?P until no augmenting path
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Augmenting path in bipartite graphs
augmenting path has odd length ? endpoints on
different sides
call the one on the left start non matching
edges
matching edges
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Augmenting path in bipartite graphs
augmenting path has odd length ? endpoints on
different sides
call the one on the left start non matching
edges
matching edges
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Augmenting path in bipartite graphs
augmenting path has odd length ? endpoints on
different sides
call the one on the left start non matching
edges
matching edges
Theorem augmenting path
exists ?
there is a path from unmatched vertex on the
left to an unmatched vertex on the right
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Augmenting path in bipartite graphs
augmenting path has odd length ? endpoints on
different sides
call the one on the left start non matching
edges
matching edges
Theorem augmenting path
exists ?
there is a path from the blue vertex on the
left to the blue vertex on the right
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Augmenting path in bipartite graphs
Finding augmenting path (or checking none
exists) one BFS (or DFS) computation TIME O(E)
M?0 repeat find augmenting path P M ?
M?P until no augmenting path
Finding max-matching O( VE )
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Maximum-weight matchings
5
5
10
10
5
5
5
5
10
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Maximum-weight matchings
5
5
10
10
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weight 30
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5
5
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maximum-weight matching doesnt have to be
maximum cardinality !!!
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Augmenting path

-


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path connecting two unmatched vertices with
alternating non-matching and matching edges
must gain by the swap
? we ? wf
f? P?M
e? P?MC
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Augmenting greedily
THEOREM
Assume M matching of the maximum-weight
among matchings with k edges P
augmenting path with maximum gain Then
M?P matching of the maximum
weight among matchings with k1
edges
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Augmenting greedily
Assume M matching of the maximum-weight
among matchings with k edges P
augmenting path with maximum gain Then
M?P matching of the maximum
weight among matchings with k1
edges
Proof B max-weight with k1 edges M?B
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Augmenting greedily
Assume M matching of the maximum-weight
among matchings with k edges P
augmenting path with maximum gain Then
M?P matching of the maximum
weight among matchings with k1
edges
gain 0
Proof B max-weight with k1 edges M?B
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Augmenting greedily
Assume M matching of the maximum-weight
among matchings with k edges P
augmenting path with maximum gain Then
M?P matching of the maximum
weight among matchings with k1
edges
gain 0
Proof B max-weight with k1 edges M?B
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Augmenting greedily
Assume M matching of the maximum-weight
among matchings with k edges P
augmenting path with maximum gain Then
M?P matching of the maximum
weight among matchings with k1
edges
gain 0
Proof B max-weight with k1 edges M?B
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Augmenting greedily
Assume M matching of the maximum-weight
among matchings with k edges P
augmenting path with maximum gain Then
M?P matching of the maximum
weight among matchings with k1
edges
Proof B max-weight with k1 edges M?B
gain wt(B) wt(M)
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How to find maximum-weight matching?
M?0 repeat find augmenting path P with
maximum gain M ? M?P until no augmenting path
only need O(V) iterations
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How to find augmenting path with the maximum gain?

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How to find augmenting path with the maximum gain?

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SHORTEST PATH PROBLEM
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How to find augmenting path with the maximum gain?
NEGATIVE CYCLE?

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How to find augmenting path with the maximum gain?
NEGATIVE CYCLE?
impossible would contradict maximality (among
matchings of cardinality k)

-
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How to find augmenting path with the maximum gain?
no negative cycles ? can use Bellman-Ford to
find the shortest path (and hence get augmenting
path with the maximum gain)
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Putting it all together
M?0 repeat find augmenting path P with
maximum gain M ? M?P until no augmenting path
only need O(V) iterations O(VE) time per
iteration (Bellman-Ford) O(V2 E) time total
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Putting it all together
M?0 repeat find augmenting path P with
maximum gain M ? M?P until no augmenting path
only need O(V) iterations O(Vlog V E) time per
iteration (Dijkstra) O(V2log V VE) time total
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Linear ordering of vertices of a directed
acyclic graph such that no edges point backward.
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2log n n
1
n1/log n (2log n )1/log n 2 (log n)/log n 2
n2/log n (2log n )2/log n 2 2(log n)/log n 4
n
n1 1/log n n . n1/log n 2n
n2 binomial(n,2)n(n-1)/2
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Mergesort reduces an instance of size n to 2
instances of size n/2 and the reduction takes
time ?(n).
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Assume that all degrees ? 2. Then 2 E ?
deg(v) ? ? 2 2 n
v?V
v?V
Hence there is a vertex of degree ? 1.
There cannot be a vertex of degree 0, since G is
connected and V?2.
Hence there is a vertex of degree 1.
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max E i , O j 1
max O i , E j 1
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max Bi-1, Bi-2 ai
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for i from 1 to n do for j from 1 to n do
Ai,j ? (Ai,j (ij)) mod 2 M1 ? largest
all-ones square M2 ? largest all-zeros
square return max (M1, M2)
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for i from 1 to n do for j from 1 to n do
Mi,j ? 1 if i1 and j1 then
if (Ai,jAi-1,j-1) and (Ai,j? Ai-1,j)
and (Ai,j? Ai,j-1)
Mi,j? 1min(Mi-1,j,Mi,j-1,Mi-1,j-1)
return max Mi,j
i,j? 1,...,n
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H heap n size of the heap
Extract-Min(H) min ? H1 H1 ? Hn n
? n-1 Heapify(H,1) return min
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input weighted, undirected graph G
(given by adjacency list)
output minimum-cost spanning tree
of G (cost of a tree is the sum of
the weights of its edges)