ALGORITHM NORTHERN LIGHTS YUN LIU DONGFENG YU TIANXIANG GAO - - PowerPoint PPT Presentation

KARGER'S MINCUT ALGORITHM

NORTHERN LIGHTS

YUN LIU DONGFENG YU TIANXIANG GAO ZHIYUAN LI XIAOYANG ZHENG

SOME SOLUTIONS

• For all pairs of vertexes u and v, run maximum flow.
• toooo slow
• Stoer–Wagner Algorithm
• nice deterministic algorithm
• Karger's Algorithm
• nice randomized alogorithm

GLOBAL MINIMUM CUT

• A cut (S,T) in an undirected graph

G=(V,E) is a partition of the vertices V into two non-empty, disjoint sets S∪T=V.

• The weight of a cut is the sum of the

weight of all edges between S and T.

• We want to find a cut with the minimum

weight among all of the cuts.

GLOBAL MINIMUM CUT

A CUT WITH VALUE 3 A MINCUT WITH VALUE 2

NOTATION

• n : number of vertexes
• m : number of edges
• degree(u) denotes the number of edges touching node u
• G/e is the graph after G contract e

KARGER’S ALGO IN UNWEIGHTED GRAPH

• The minimum cut is usually very small. So if we randomly

choose an edge, it is very likely to be not in the mincut.

KARGER’S ALGO IN UNWEIGHTED GRAPH

• 1. Pick an edge of G at random and collapse its two endpoints

into a single node.

• 2. Repeat 1 until just two nodes remain.
• The edges between the two nodes is a cut.
• We claim this procedure is very likely to find a mincut.

Repeat until just two nodes remain:

KARGER’S ALGO IN UNWEIGHTED GRAPH

• 1. Pick an edge of G at random and collapse its two endpoints

into a single node.

• 2. Repeat 1 until just two nodes remain.
• The edges between the two nodes is a cut.
• We claim this procedure is very likely to find a mincut.

Repeat until just two nodes remain:

KARGER’S ALGO IN UNWEIGHTED GRAPH

• 1. Pick an edge of G at random and collapse its two endpoints

into a single node.

• 2. Repeat 1 until just two nodes remain.
• The edges between the two nodes is a cut.
• We claim this procedure is very likely to find a mincut.

Repeat until just two nodes remain:

KARGER’S ALGO IN UNWEIGHTED GRAPH

• 1. Pick an edge of G at random and collapse its two endpoints

into a single node.

• 2. Repeat 1 until just two nodes remain.
• The edges between the two nodes is a cut.
• We claim this procedure is very likely to find a mincut.

Repeat until just two nodes remain:

KARGER’S ALGO IN UNWEIGHTED GRAPH

• 1. Pick an edge of G at random and collapse its two endpoints

into a single node.

• 2. Repeat 1 until just two nodes remain.
• The edges between the two nodes is a cut.
• We claim this procedure is very likely to find a mincut.

Repeat until just two nodes remain:

KARGER’S ALGO IN UNWEIGHTED GRAPH

• 1. Pick an edge of G at random and collapse its two endpoints

into a single node.

• 2. Repeat 1 until just two nodes remain.
• The edges between the two nodes is a cut.
• We claim this procedure is very likely to find a mincut.

Repeat until just two nodes remain:

KARGER’S ALGO IN UNWEIGHTED GRAPH

• 1. Pick an edge of G at random and collapse its two endpoints

into a single node.

• 2. Repeat 1 until just two nodes remain.
• The edges between the two nodes is a cut.
• We claim this procedure is very likely to find a mincut.

Repeat until just two nodes remain:

KARGER’S ALGO IN UNWEIGHTED GRAPH

• 1. Pick an edge of G at random and collapse its two endpoints

into a single node.

• 2. Repeat 1 until just two nodes remain.
• The edges between the two nodes is a cut.
• We claim this procedure is very likely to find a mincut.

Repeat until just two nodes remain:

A FEW LAMMAS

• A cut in G/e is also a cut in G

A FEW LAMMAS

• Let S and T be a mincut, suppose e does not cross the cut,

mincut(G) = mincut(G/e)

P(FOUND A MINCUT)≥

2 𝑜2

• Prove that there is a vertex with degree not greater than 2 𝑛

𝑜.

• Proof
• The sum of degree is 2m since each edge has 2 ends. So, the

average vertex degree is 2 𝑛

𝑜. Therefore, there exists at least

• ne vertex with degree not greater than 2 𝑛

𝑜.

P(FOUND A MINCUT)≥

2 𝑜2

• Prove that the size of the mincut is at most 2

𝑛 𝑜.

• Proof
• Consider the partition of G into two pieces, one

containing a single node u, and the other containing the remaining n − 1 nodes. The size of this cut is degree(u). Since we already prove there is a vertex with degree not greater than 2

𝑛 𝑜 , there exists a cut with value at most

2

𝑛 𝑜. So, that the size of the mincut is at most 2 𝑛 𝑜.

P(FOUND A MINCUT)≥

2 𝑜2

• Prove that for a randomly chosen edge e, P(e is not in the

mincut)≥

𝑜−2 𝑜

• Proof
• The size of the mincut is at most 2

𝑛 𝑜, so P(e is in the

mincut)≤

2𝑛 𝑜 × 1 𝑛 = 2 𝑜.

• Therefore, P(e is not in the mincut)≥ 1 −

2 𝑜 = 𝑜−2 𝑜

P(FOUND A MINCUT)≥

2 𝑜2

• Prove that P(found a mincut)≥ 2

𝑜2

• Proof
• Pr(final cut is the minimum cut) = Pr(first selected edge is not in

mincut) × Pr(second selected edge is not in mincut) × · · · = 𝑜−2

𝑜 × 𝑜−3 𝑜−1 × 𝑜−4 𝑜−2 … × 2 4 × 1 3 = 2 𝑜(𝑜−1) ≥ 2 𝑜2

REPEAT IT 𝑂2 𝑀𝑃𝐻 𝑂 TIMES

• Prove that P(found a mincut after 𝑜2 log 𝑜 times)≥ 1 − 1

𝑜

• Proof
• P(found a mincut after 𝑜2 log 𝑜 times)≥ 1 − 1 − 1

𝑜2 2 𝑜2 2 ×ln 𝑜 ≥

1 − 1 − 1 𝑓ln 𝑜 = 1 − 1 𝑜

TIME COMPLEXITY

• 𝑜2 ⋅ 𝑜2 log 𝑜 = 𝑜4 log 𝑜

A STRONG IMPROVEMENT

• This toooo slow.
• P(e is not in the mincut)≥ 1 −

2 𝑜 is almost 1 when n is

large, but decreases to

1 3 when n=3.

• We want to avoid that.

A STRONG IMPROVEMENT

A STRONG IMPROVEMENT

TIME COMPLEXITY

• 𝑜2 log 𝑜 ⋅ log2 𝑜 = 𝑜2 log3 𝑜

WE CAN DO IT IN WEIGHTED GRAPH

• 𝑃(𝑜2 log3 𝑜)

SUMMARY

• Karger’s Algorithm in unweighted Graph

𝑃(𝑜4𝑚𝑝𝑕𝑜)

• A strong improvement

𝑃(𝑜2𝑚𝑝𝑕3𝑜)

• Karger’s Algorithm in weighted Graph

𝑃(𝑜2𝑚𝑝𝑕3𝑜)

REFERENCES

[1] D. R. Karger and C. Stein. A new approach to the minimum cut problem. J. ACM, 43(4):601–640, 1996.