On the uniform edge-On the uniform edge-partition of a treepartition of a tree

吳邦一 樹德科大 資工系吳邦一 樹德科大 資工系王弘倫 台大 資工系王弘倫 台大 資工系管世達 樹德科大 資工系管世達 樹德科大 資工系趙坤茂 台大 資工系趙坤茂 台大 資工系

vertex partition of a vertex partition of a treetree

2-partition3-partition

Tree splitting Tree splitting (edge partition)(edge partition)

2-split3-split

Objective functionsObjective functions

min-max max-min minimize largest

smallest

Previous resultsPrevious results

tree vertex partition: (weighted)tree vertex partition: (weighted)– min-max or max-min: polynomial timemin-max or max-min: polynomial time– most-uniform: unknownmost-uniform: unknown

For a path and the objective is to minimize the differFor a path and the objective is to minimize the difference: polynomial time.ence: polynomial time.

The most uniform partition:The most uniform partition:– No report (to our best knowledge) even for set No report (to our best knowledge) even for set

partition.partition.– tree splitting: apparently NP-hard (3-partition) tree splitting: apparently NP-hard (3-partition)

even for unweighted edges.even for unweighted edges.

Our resultsOur results

The tree The tree kk-splitting is -splitting is NPNP-hard.-hard. For For kk 4, the existence of a 4, the existence of a kk - -

splitting for any tree with ratio at splitting for any tree with ratio at most.most.– a 2-approximation algorithma 2-approximation algorithm

A simple 3-approximation A simple 3-approximation algorithm for general algorithm for general k. k. – Experimental results included. Experimental results included.

A simple property A simple property

For any 1 For any 1 ee((TT), we can split ), we can split TT in into (to (TT11, , TT22) at a vertex ) at a vertex vv in linear time s in linear time such that uch that ee((TT11) ) 2 2..

YY

Y

each y each y Corollary: A tree can be spit into T1 and T2, n/3 ee((TT1) , ) , ee((TT2) ) 2n/3

For k = 3 For k = 3

n/4 n/4 y x n/2

YP0 X

n/4 n/4 y y n/2n/2

n/4 n/4 x x n/2n/2

Two casesTwo cases

y y 2n/5 2n/5 < y x n/2

Case 1: n/4 Case 1: n/4 y 2n/5

YP0 X

T1P1

P 2

n/4n/4 yy 2n/52n/5 PP11 2T2T11/3/3 n/2n/2

PP2 2 T T11/3 /3 n/4n/4

Case 2: Case 2: 2n/5 < y x n/2

YP0 X X 1X

2

n/5 n/5 X X11 2n/52n/5

XX11XX22

Only need to consider Only need to consider n/5 n/5 x x11 < n/4 < n/4

2n/5 < y n/2, y/2 y/2 x x11 < y < y n/4 < n/4 < n-xn-x11-y-y< 2n/5< 2n/5 (X(X11, X, X22PP00, Y) is a desired splitting, Y) is a desired splitting

YP0 X 1X

2e(X2e(X2P0)P0)

For k=4For k=4

It can be prove in a similar way, It can be prove in a similar way, but the cases are more but the cases are more complicated.complicated.

A simple algorithmA simple algorithm

There is a simple algorithm to There is a simple algorithm to split a tree with ratio at most 3.split a tree with ratio at most 3.

Method: always split the Method: always split the maximum part of the previous maximum part of the previous splitting.splitting.

2e2e ee3e3e

Proof:Proof:

By induction.By induction.

ee 3e3e

2e2e

ratio3

ratio3

Experimental resultExperimental result

Thank youThank you