Problem with Heuristic Search YOSSI COHEN P R O F . A R I E L F E - - PowerPoint PPT Presentation

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Problem with Heuristic Search YOSSI COHEN P R O F . A R I E L F E - - PowerPoint PPT Presentation

Jul 17, 2023 501 likes β€’752 views

Solving the Longest Simple Path Problem with Heuristic Search YOSSI COHEN P R O F . A R I E L F E L N E R , D R . R O N I S T E R N 1 LSP - Definition s # # # # # #

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YOSSI COHEN

P R O F . A R I E L F E L N E R , D R . R O N I S T E R N

Solving the Longest Simple Path Problem with Heuristic Search

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LSP - Definition

ο‚— Find the longest simple path from Start to Target (𝑑 β†’ 𝑒). ο‚— NP Hard problem - even hard to estimate by a constant factor s # # # # # # # # # # # # # # # # # # # # t ↓ β†’ β†’ ↓ # # β†’ β†’ β†’ ↓ ↓ ↑ # β†’ β†’ ↓ ↑ ↓ ← ↓ ↑ ← ← # β†’ ↑ # β†’ ↓ β†’ β†’ β†’ ↑ # # ↓ ← ↓ # # ↓ ← # ↓ ↑ ← ↓ ← ← ← ↑ ← ↓ ← # ↓ # # # ↑ ← # # β†’ β†’ ↓ β†’ β†’ ↓ β†’ ↓ # # ↓ ↑ ← β†’ ↑ ↓ β†’ ↓ # β†’ β†’ ↑ # β†’ ↑ t

Background

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Motivation

Real world application:

ο‚— VLSI design (integrated circuit design) ο‚— Gray code - error correction ο‚— Robot patrolling

Background

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ο‚— Unlike many problems that are solved by finding a

solution with minimal cost, LSP solution requires maximal reward.

ο‚— Simple Path from 1 to 2 ο‚— MIN vs MAX ο‚— Shortest Path ο‚— Longest Simple Path (LPP)

Maximum vs. Minimum

Background

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What’s the problem? Lets just use Best First Search!

ο‚— [s:0] ο‚— [a:1, b:2] ο‚— [a:1,g:3]

MIN vs. MAX

s g a b 2 1 1 3

Background

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Search Space is Bigger

Both nodes are same place - but they not the same valid operators: [S,W] [N,S] Moreover one is part of MAX solution and the other is not. We must keep the entire path in every search node. Due to this the search space is much bigger.

t s t s

Background

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ο‚— f(Β·) = g(Β·) + h(Β·) ο‚— A*

ο‚‘ MAX problems: the first state in the open list is the maximal

  • state. (denote maxf)

ο‚‘ Stops when open-list is empty or when:

maxf ≀ BestGoalFound

ο‚— DFBnB

ο‚‘ If F < best candidate – prune

ο‚— Better heuristics are heuristics that tightly upper

bounds the remaining path

A* and DFBnB Adaptations

Background

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Heuristics Shortest path heuristics just won't work here

Heuristics

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Reachable Biconnected components

Existing heuristics

Heuristics

s t f Cut point Block s t t s

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Works on bipartite graph. 4 connected grid for instance. Count the groups separately, |Ξ”| ≀ 1

Alternate Steps

Heuristics

S # # # T S # # # T S # # # T

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Run Alternate step (Alt.) on each block of the BCT separately. h(BCC alt.) = 33 h(BCC s. alt.) = 31

BCC + Separate Alternate Steps

Heuristics

S # # # # # # # T

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Pruning

How to prune nodes during the search and still guarantee to find the longest simple path?

Pruning

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Search Tree Example

t s t s t s t s t s t s t s t s t s t s

Pruning

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Search Tree Example

t s t s t s t s t s t s t s t s t s t s

When using heuristic search to solve LSP A* traverse over many similar states

s t t s f f

Pruning

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Basic Symmetry Detection

Same frontier location & same path coverage On open list: Generated node:

ο‚— How to efficiently compare states?

s s t t f f

Pruning

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Reachable Dominance Detection

Same frontier & contained (subset or equal) reachable coverage On open list: Generated node:

ο‚— Can prune retroactively! ο‚— How to efficiently compare states?

s t s t f f

Pruning

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Reachable Dominance Detection

s t s t f f

Pruning

Pruning Conditions:

  • 1. N.head = Nβ€˜.head

2.|N.Ο€|β‰₯|Nβ€˜.Ο€| 3.Nβ€˜.RβŠ† N.R N Nβ€˜

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Experimental Results

Experimental Results

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Experimental Results

ο‚— Grid maps with random blocked cells - 360 maps

with variety of blocked percentile

ο‚— 10Minutes, solved by all permutations

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Experimental Results

ο‚— Grid maps with rooms - 400 maps with variety of

number of rooms, room size and blocked percentile

ο‚— 10Minutes, solved by all permutations

A* DFBnB Runtime

(BCC Based only)

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Experimental Results

Success rate – Bigger grids and non-uniform reward 1Hr.

Uniform reward Life Grid (non-uniform)

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Conclusion

Contributions

ο‚— Novel heuristic for the longest simple path problem ο‚— Several state space pruning techniques

Results

ο‚— All proposed pruning techniques reduce # searched nodes. ο‚— Pruning effectiveness: None ≀ BSD ≀ RDP ο‚— Heuristic effectiveness: R ≀ R+ALT ≀ BCC ≀ BCC+ALT ≀ BCC+Sep. ALT

Challenge for Future Work

ο‚— BSD has the fastest runtime and RDD has the strongest pruning ability ο‚— Challenge: How to get RDD faster?

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Thanks!

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https://github.com/YossiCohen/Heuristic-Search-Max