Constructing Minimal Spanning Steiner Trees with Bounded Path LengthPresenter : Cheng-Yin Wu, NTUGIEE

Constructing Minimal Spanning Steiner Trees with Bounded

Path Length

Some of the Slides in this Presentation are Referenced from : Physical Design for Nanometer IC by Prof. Yao-Wen Chang

• A real world critical problem

– (Clock Tree) Routing in VLSI Circuit

Introduction to “Bounded” Tree (1/2)

Source : Physical Design for Nanometer IC by Prof. Yao-Wen Chang

• Critical Path Delay (Speed)

– Maximum delay from source to any sink

– Optimum : A shortest path tree rooted at source

• Routing Capacitance (Power)

– Power dissipation is linear to routing capacitance – Optimum : A minimum spanning tree

• Draw a balance between : Speed and Power

– Balance between MST and SPT ?!

Introduction to “Bounded” Tree (2/2)

MST v.s. SPT Trade-off – Example

Cost (MST) ≤ R + π Cost (SPT) = (N – 1)R Source (s)

Sinks

• Define R = max dist

(s, v ϵ V)

• Define Cost(T) = ∑ w(e ϵ T(E))

Bounded Radius Spanning Tree

• Known as Bounded path length minimum spanning tree

• radius

(u) = max dist

(u, v ϵ V)

– R = radius_T (s) if T = SPT(s)

• Problem Formulation

– Given routing graph G(V, E) with w(e ϵ E) > 0, find a minimal cost routing tree T with

radius_T (s) ≤ (1 + ε) R for any given ε ≥ 0

– Two extreme cases : MST(ε  ∞) and SPT(ε = 0)

NP-Complete

Overview of My Following Presentation

• Both heuristics and exact algorithms

• Heuristics (Sound Not Complete)

– Previous works : BPRIM, BRBC – This work : BKRUS, BH2

• Exact Algorithms (Sound and Complete)

– (Improved) Previous works : BMST_G – This work : BKEX

• To the best of my knowledge, it is the most premier work in tackling this problem

BRBC BKRUS BH2

BKEX

Bounded Radius Bounded Cost Spanning Tree (1/2)

Awerbuch, et al., “Cost-sensitive analysis of communication protocols,” ACM Symp. Principles of Distributed Computing, 1990.

Cong, Kahng, Robins, Sarrafzadeh, Wong, “Performance-driven global routing for cell based IC's,” ICCD-91 (and TCAD, June 1992)

• One of the first work on this problem

• Claims

– radius_T (s) ≤ (1 + ε) R

– Cost(T) ≤ (1 + 2/ε) Cost(MST) (Omit Today) – Polynomial time Heuristic

Bounded Radius Bounded Cost Spanning Tree (2/2)

Source : Physical Design for Nanometer IC by Prof. Yao-Wen Chang

BKRUS : Bounded Kruskal MST

• A heuristic similar to BPRIM (previous work), BKRUS extends existing Kruskal MST algorithm to bounded radius cases

– Achieving better performance over benchmarks

• Goal : radius

(s) ≤ (1 + ε) R

• Classical Kruskal MST Algorithm – Review

– http://www.cs.man.ac.uk/~graham/cs2022/greedy/

BKRUS Algorithm (1/3)

• Extension for upper bound constraint

– Two subtrees t_u, t_v are merged by edge (u, v) if the merged tree t_M satisfies radius_tM (s) ≤ (1 + ε) R

u

t

t

t

u

t

t

BKRUS Algorithm (1/3)

• Extension for upper bound constraint

– Two subtrees t_u, t_v are merged by edge (u, v) if the merged tree t_M satisfies radius_tM (s) ≤ (1 + ε) R

– Case 1 : If s ϵ t_u

dist_tu(S, u) + dist_G(u, v) + radius_tv(v) ≤ (1 + ε) R

– Similar case for s ϵ t_v

BKRUS Algorithm (1/3)

• Extension for upper bound constraint

– Two subtrees t_u, t_v are merged by edge (u, v) if the merged tree t_M satisfies radius_tM (s) ≤ (1 + ε) R

– Case 2 : If neither s ϵ t_u nor s ϵ t_v dist_G(S, x) + radius_tM(x) ≤ (1 + ε) R

– If such x does not exist (u, v) is not feasible

u

t

t

x

BKRUS Algorithm (2/3)

• BKRUS maintains two matrices for t

– P[V, V] denotes path lengths between (V, V) pairs – r[V] denotes radii of every V

(1). If s ϵ t_u, t_v, check : dist_tu(S, u) + dist_G(u, v) + radius_tv(v) ≤ (1 + ε) R (2). Else, check : dist (S, x) + radius (x) ≤ (1 + ε) R, x ϵ t

BKRUS Algorithm (2/3)

BKRUS Algorithm (3/3)

O(V) O(V²) ONLY (V – 1) times

E times

Complexity = O(EV + V³)

Extension to Elmore Delay Model

• Elmore delay model is a more accurate delay estimation model on VLSI circuit

– Now, path delay is a function to driving

network, not edge itself

• So, new radii after each MERGE should be re- computed again

• dist(x, y) is replaced by R(x, y)(C(x, y)/2 + C

)

BKRUS Algorithm for Steiner Tree

• A variation of BKRUS, named BKST

Hanan Grid

Exact Algorithms – Gabow

• Definition : T-exchange

– Given a pair of edges (e, f) where e ϵ T, f ϵ G \ T, and T \ e U f is a spanning tree, we define

exchange of (e, f) has weight = w(f) – w(e)

• Gabow’s Algorithm tries to find k smallest

spanning trees evolved from original spanning tree with one exchange in every iteration

– Problem : Exponential Space Complexity (V^{V – 2})

– Solution : Once a feasible solution is found, return

Exact Algorithms – Gabow

• More on T-exchange

– If T is the minimum spanning tree, i.e. Cost(T) is the minimum among all spanning trees

• NO negative weight of exchange can be found

– Further, if (e, f) is the minimal T-exchange, T \ e U f is a spanning tree with the next smallest cost

• In reality, some edges in original graph can be excluded at a preprocessing of Gabow’s

algorithm (omitted here)

Exact Algorithms – BKEX (1/2)

• Motivation : Global Optimization

• Definition : Negative-Sum-Exchange(s)

– A sequence of T-exchanges where sum of their weights are negative

• BKEX starts from initial feasible solution, and reduce routing cost toward the optimal

– As a post-processing algorithm for, e.g. BKT

Exact Algorithms – BKEX (2/2)

Complexity = O(Eⁿ Vⁿ⁺¹)

depth = 0 depth = 1 depth = 2

depth = 3

depth = 4

depth = 5

y x

Heuristic from Exact Algorithms – BH2

• BKT is a local minimum solution

– If an edge is rejected during the BKRUS algorithm, then that edge cannot become feasible at a later time (proof is omitted in the presentation)

– It can be shown BKT is a local optimum with respect to a single T-exchange

• In order to jump out of local optimum, at least double T-exchanges are needed  BH2

– E.g. negative-sum-exchanges : +3, -5  -2 – Complexity = O(E² V³)

Upper and Lower Bounded MST

• Sometimes we desire a spanning tree that consider both max and min dist

(S, x) while minimizing routing cost

– For instance, clock skew in clock tree synthesis

• Problem can be re-formulated as

– ε₁ R ≤ radius_T (s) ≤ (1 + ε₂)

– Some combinations of ε₁, ε₂ render NO solution

• However, for Steiner Trees, solutions may exist

• How can BKRUS be extended now ?!

Experiments – Tradeoff Curve

Experiments – BH2 Improvements

Experiments – Routing Cost Over MST

– Max = 2.711, 2.219, 2.056, 2.056 – Avg = 1.705, 1.517, 1.455, 1.452

Conclusion and Future Work

• Schemes for bounded path length minimal spanning tree with upper (and lower) bound control are presented

– With applications to , e.g. Elmore delay model

• Two optimal algorithms

– BMST_G extends Gabow’s method

– BKEX using negative T-exchange technique

• Heuristic BKRUS can be further enhanced with

BKH2 as a post processing