Ecient nondestructive equality checking for trees and graphs
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1 Ecient nondestructive equality checking for trees and graphs Michael D. Adams R. Kent Dybvig ICFP, 2008
2 Overview 1 Background 2 Basic Implementations R5RS Implementation Union-nd implementation 3 Optimized Implementations Precheck Interleaved Interleaved with Precheck 4 Summary Adams (Indiana Univ.) Nondestructive equality ICFP / 30
3 Background Revised 5 Report on the Algorithmic Language Scheme (R5RS): Equal? may fail to terminate if its arguments are circular data structures Adams (Indiana Univ.) Nondestructive equality ICFP / 30
4 Background Scheme Request for Implementation 85 (SRFI-85): Denes equiv? Terminates on all input Reference implementation Adams (Indiana Univ.) Nondestructive equality ICFP / 30
5 Background Revised 6 Report on the Algorithmic Language Scheme (R6RS): The equal? predicate returns #t if and only if the (possibly innite) unfoldings of its arguments into regular trees are equal as ordered trees. Adams (Indiana Univ.) Nondestructive equality ICFP / 30
6 Requirements Correct Unknown inputs Not slow Adams (Indiana Univ.) Nondestructive equality ICFP / 30
7 R5RS (Tree) Equality
8 R5RS (Tree) Equality (dene (equal? x y) (cond [(eq? x y) #t] [(pair? x) (and (pair? y) (equal? (car x) (car y)) (equal? (cdr x) (cdr y)))] [(vector? x) (and (vector? y)...)]...)) Adams (Indiana Univ.) Nondestructive equality ICFP / 30
9 R5RS (Tree) Equality On non-sharing: Fast Adams (Indiana Univ.) Nondestructive equality ICFP / 30
10 R5RS (Tree) Equality On non-sharing: Fast On cycles: Loops forever Adams (Indiana Univ.) Nondestructive equality ICFP / 30
11 R5RS (Tree) Equality On non-sharing: Fast On cycles: Loops forever On acyclic sharing:. Exponential Adams (Indiana Univ.) Nondestructive equality ICFP / 30
12 Union-Find (Graph) Equality
13 Union-Find: Algorithm Equivalence classes Each call Trivial Same equivalence class Dierent equivalence classes Adams (Indiana Univ.) Nondestructive equality ICFP / 30
14 Union-Find Adams (Indiana Univ.) Nondestructive equality ICFP / 30
15 Union-Find: Analysis Union-nd Equality Inverse Ackermann (almost constant) Almost linear DFA equivalence algorithms Adams (Indiana Univ.) Nondestructive equality ICFP / 30
16 Union-Find: Implementation Pointer per object Store in object header Not pure/functional Overhead to every object Threads Adams (Indiana Univ.) Nondestructive equality ICFP / 30
17 Union-Find: Implementation Pointer per object Store in object header Not pure/functional Overhead to every object Threads Use eq hash tables Map object to pointer GC Interaction Adams (Indiana Univ.) Nondestructive equality ICFP / 30
18 Union-Find: Performance 25x slower than tree-equality Hash table Union-nd algorithm Adams (Indiana Univ.) Nondestructive equality ICFP / 30
19 Precheck
20 Precheck Algorithm Tree-equality for N steps After N steps, restart and run union-nd Basic idea in SRFI-85 Adams (Indiana Univ.) Nondestructive equality ICFP / 30
21 Setting Precheck Bound Bound too big Bound too small No perfect bound Adams (Indiana Univ.) Nondestructive equality ICFP / 30
22 Interleaved Equality
23 Interleaved Want Idea Fast tree-equal on trees Required union-nd-equal on cycles/dags Co-routines Adams (Indiana Univ.) Nondestructive equality ICFP / 30
24 Interleaved Tree-equality for k 0 steps Union-nd for k b steps But continue if successful Adams (Indiana Univ.) Nondestructive equality ICFP / 30
25 Interleaved Tree-equality for k 0 steps Union-nd for k b steps Synergy But continue if successful Tree parts get tree-equality Cycle and graph parts get union-nd Adams (Indiana Univ.) Nondestructive equality ICFP / 30
26 Analytic Performance Within constant of union-nd Start with union-nd k 0 /k b Start with tree mode k 0 Adams (Indiana Univ.) Nondestructive equality ICFP / 30
27 Where do we start? Union-nd Tree Pro: k 0 /k b, large cycles Con: small trees Pro: small trees Con: k 0, large cycles Adams (Indiana Univ.) Nondestructive equality ICFP / 30
28 Where do we start? Union-nd Tree Pro: k 0 /k b, large cycles Con: small trees Pro: small trees Con: k 0, large cycles Neither solution acceptable Adams (Indiana Univ.) Nondestructive equality ICFP / 30
29 Interleaved with Precheck
30 Interleaved with Precheck Interleave and precheck Small trees Large cycles Adams (Indiana Univ.) Nondestructive equality ICFP / 30
31 Benchmarks Lists Rev. lists Rand. DAGS Balanced binary tree Degenerate DAGS Random graph Union-exercising graph Adams (Indiana Univ.) Nondestructive equality ICFP / 30
32 Benchmark r5rs uf precheck/uf interleave precheck/interleave Adams (Indiana Univ.) Nondestructive equality ICFP / 30
33 Conclusions Simple idea, subtle performance implications Mixing best of all worlds Tree Union-nd Interleaving Precheck Adams (Indiana Univ.) Nondestructive equality ICFP / 30
34 Conclusions Interleaving with precheck Never the best Always almost the best Particular applications vs. General purpose library Adams (Indiana Univ.) Nondestructive equality ICFP / 30
35 Questions?
Efficient Nondestructive Equality Checking for Trees and Graphs
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