Average Case Lower Bounds for Monotone Switching Networks
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1 Average Cae Lower Bound for Monoone Swiching Nework Yuval Filmu, Toniann Piai, Rober Robere, Sephen Cook Deparmen of Compuer Science Univeriy of Torono
2 Monoone Compuaion (Refreher) Monoone circui were inroduced a a naural rericion on general circui for he purpoe of proving lower bound.
3 Monoone Compuaion (Refreher) Monoone circui were inroduced a a naural rericion on general circui for he purpoe of proving lower bound. Many of he (general) circui cla eparaion we believe o be rue imply heir monoone verion, o monoone eparaion are a good place o ar.
4 Monoone Compuaion (Refreher) Monoone circui were inroduced a a naural rericion on general circui for he purpoe of proving lower bound. Many of he (general) circui cla eparaion we believe o be rue imply heir monoone verion, o monoone eparaion are a good place o ar. Much of he work up unil now ha been on exac monoone compuaion --- wha abou average-cae?
5 Previou Monoone Lower Bound Problem Clique Direced Conn. Poiner Jumping GEN GEN
6 Previou Monoone Lower Bound Problem Clique Direced Conn. Poiner Jumping GEN GEN Fir Proved By... Razborov [85] Karchmer-Wigderon [88] Grigni-Siper [95] Raz-McKenzie [97] Chan-Poechin [12]
![Grigni-Siper [95] Raz-McKenzie [97]](/docs-images/88/115131545/images/7-5.jpg "Chan-Poechin [12] Lower Bound (Ck Size)")
7 Previou Monoone Lower Bound Problem Clique Direced Conn. Poiner Jumping GEN GEN Fir Proved By... Razborov [85] Karchmer-Wigderon [88] Grigni-Siper [95] Raz-McKenzie [97] Chan-Poechin [12] Lower Bound (Ck Size) (Ck Deph) (Ck Deph) (Ck Deph) (Nework Size)
8 Monoone, Average Cae LB Problem Clique Diribuion Random Graph Lower Bound (Roman [10])
9 Monoone, Average Cae LB Problem Clique Direced Conn. Diribuion Random Graph Lower Bound (Roman [10]) Poiner Jumping GEN??
10 Monoone, Average Cae LB Problem Clique Direced Conn. Diribuion Random Graph Lower Bound (Roman [10]) Poiner Jumping GEN?? Our work
11 Monoone Swiching Nework Model for ml baed on undireced conneciviy x z z y v z
12 Monoone Swiching Nework Model for ml baed on undireced conneciviy x z z y v z x = z = 1 y = v = 0
13 Monoone Swiching Nework Model for ml baed on undireced conneciviy x z z y v z x = z = 1 y = v = 0 Dead Alive
14 Monoone Swiching Nework Model for ml baed on undireced conneciviy x z z y v z x = z = 1 y = v = 0 - live pah, o accep!
15 Monoone Swiching Nework Model for ml baed on undireced conneciviy x z z y v z y = z = 1 x = v = 0 Alo accep!
16 Monoone Swiching Nework Model for ml baed on undireced conneciviy x z z y v z x = z = v = 1 y = 0 Two acceping pah, bu ha alrigh!
17 Monoone Swiching Nework y x z v z z Allow reverible nondeerminim
18 Monoone Swiching Nework y x z z z v M compue a boolean funcion f if f(x) = 1 iff M(x) ha a live - pah
19 Monoone Swiching Nework y x z v z z ml = all problem olvable by polynomially-ized monoone wiching nework
20 Monoone Swiching Nework In hi model, he canonical complee problem for ml i undireced graph conneciviy:
21 Monoone Swiching Nework In hi model, he canonical complee problem for ml i undireced graph conneciviy: u Given a li of edge in an undireced graph, i here an - pah? v
22 Monoone Swiching Nework In hi model, he canonical complee problem for ml i undireced graph conneciviy: u u u uv v v v
23 Monoone Swiching Nework In hi model, he canonical complee problem for ml i undireced graph conneciviy: u u u uv v v v
24 Swiching Nework and Circui Swiching nework ize and circui deph are cloely relaed. Theorem: A wiching nework of ize O() compuing a boolean funcion f can be convered ino a bounded fan-in circui wih deph Theorem: A bounded fanin circui of deph O(d) compuing a boolean funcion f can be convered ino a wiching nework wih ae.
25 Swiching Nework and Circui Swiching nework ize and circui deph are cloely relaed. Theorem: A wiching nework of ize O() compuing a boolean funcion f can be convered ino a bounded fan-in circui wih deph In oher word, a lower bound of on wiching nework ize implie a deph lower Theorem: A bounded of fanin circui for circui. of deph O(d) compuing a boolean funcion f can be convered ino a wiching nework wih ae.
26 The GEN Problem Inpu: Poiive ineger N, a ube of riple and wo diinguihed poin, in [N]. Problem: Doe generae uing he riple in L?
27 The GEN Problem (Algebraic Inuiion) Inpu: Poiive ineger N, a muliplicaion able and wo diinguihed poin, in [N]. Problem: Doe lie in he e generaed by repeaed powering of?
28 Generaion uing Triple (Inuiion) Think of each riple (x, y, z) a a pair of direced edge: (x, y) and (y, z) N = 6
29 Generaion uing Triple (Inuiion) (,, a) a b N = 6
30 Generaion uing Triple (Inuiion) (, a, b) (Order of fir wo coordinae doen maer) a b N = 6
31 Generaion uing Triple To generae we ar wih, and ee wha we can reach wih he riple a b N = 6
32 Generaion uing Triple To generae we ar wih, and ee wha we can reach wih he riple a b N = 6 Uing (,,a) and {} we generae {,a}
33 Generaion uing Triple To generae we ar wih, and ee wha we can reach wih he riple a b N = 6 Uing (,a,b) and {,a} we generae {,a,b}
34 Generaion uing Triple And o on... a b N = 6
35 Generaion uing Triple If ever ener our generaed e we accep he inance! a b N = 6
36 Graph Inance Viewing GEN a a graph i a nice implificaion: If G i a DAG where each node ha fan-in 0 or 2, we aociae a naural GEN inance wih riple (x,y,z) for each pair of edge (x,z), (y,z). Graph Inance of GEN iomorphic o G
37 GEN i in mp I i eay o give a polynomially-ized monoone circui family for GEN.
38 GEN i in mp We have one wire for each poin in he inpu e [N]. A wire will have value 1 during he compuaion if ha poin can be generaed. u v
39 GEN i in mp Conruc a gadge G ha updae he conen of he wire, uing he inpu riple. u v G
40 GEN i in mp Conruc a gadge G ha updae he conen of he wire, uing he inpu riple. = 1 u = 1 v G v = 1 if v gen. from {, u} = 1 if gen. from {, u}
41 GEN i in mp How doe G work? For each poin z, e he wire of z o be equal o. = 1 u = 1 v G v = 1 if v gen. from {, u} = 1 if gen. from {, u}
42 GEN i in mp How doe G work? For each poin z, e he wire of z o be equal o. So, G ha ize (looely). = 1 u = 1 v G v = 1 if v gen. from {, u} = 1 if gen. from {, u}
43 GEN i in mp To compue GEN, we imply ring n copie of G in a row, aring wih = 1 and all oher wire wih 0. = 1 u v G G G G
44 GEN i in mp To compue GEN, we imply ring n copie of G in a row, aring wih = 1 and all oher wire wih 0. = 1 u v Wha G abou monoone G wiching G nework? G
45 Reverible Pebbling Game Given: A DAG G wih a ource node and a ink node
46 Reverible Pebbling Game Given: A DAG G wih a ource node and a ink node Goal: Place a pebble on he ink node, minimize # of pebble
47 Reverible Pebbling Game Given: A DAG G wih a ource node and a ink node Goal: Place a pebble on he ink node, minimize # of pebble One move: If all he predeceor of a node have pebble, hen place a pebble on or remove a pebble from ha node.
48 Reverible Pebbling Game, Example ha no predeceor, o we can alway place a pebble on i
49 Reverible Pebbling Game, Example
50 Reverible Pebbling Game, Example
51 Reverible Pebbling Game, Example Now remove pebble, ince we don need hem...
52 Reverible Pebbling Game, Example Now remove pebble, ince we don need hem
53 Reverible Pebbling Game, Example Noe ha if we removed he pebble on fir, we canno remove he oher pebble due o reveribiliy:
54 Reverible Pebbling Game The reverible pebbling game naurally define an efficien wiching nework algorihm for GEN on hi graph.
55 Reverible Pebbling Game The reverible pebbling game naurally define an efficien wiching nework algorihm for GEN on hi graph. a b
56 Reverible Pebbling Game The reverible pebbling game naurally define an efficien wiching nework algorihm for GEN on hi graph. No ored! a b
57 Reverible Pebbling Game The reverible pebbling game naurally define an efficien wiching nework algorihm for GEN on hi graph. We only ore c log n bi, where c i he maximum number of pebble.
58 Reverible Pebbling Game The reverible pebbling game naurally define an efficien wiching nework algorihm for GEN on hi graph. We only ore c log n bi, where c i he maximum number of pebble. Require ae
59 Reverible Pebbling Game The reverible pebbling game naurally define an efficien wiching nework algorihm for GEN on hi graph. We only ore c log n bi, where c i he maximum number of pebble. We how ha hi i eenially igh, even when he monoone algorihm i allowed o err on a large fracion of he inpu.
60 Reverible Pebble Number If G i a DAG hen define he reverible pebbling number of G o be he minimum number of pebble needed o pebble he ink of G.
61 Reverible Pebble Number If G i a DAG hen define he reverible pebbling number of G o be he minimum number of pebble needed o pebble he ink of G. (Cook [73]) Heigh h pyramid
62 Reverible Pebble Number If G i a DAG hen define he reverible pebbling number of G o be he minimum number of pebble needed o pebble he ink of G. (Cook [73]) Heigh h pyramid (Poechin [12]) Pah wih n node
63 Deerminiic Lower Bound Poechin give a mehod (refined by Chan and Poechin) o lower bound he number of ae of monoone wiching nework compuing cerain funcion.
64 Deerminiic Lower Bound Poechin give a mehod (refined by Chan and Poechin) o lower bound he number of ae of monoone wiching nework compuing cerain funcion. High level: 1. Chooe a large e of YES inance S. 2. Show ha, for any wiching nework M and any inance I in S, you can find a ae in M ha i highly pecific o I. 3. Show ha any ae can only be highly pecific o a relaively mall number of inance I.
65 Deerminiic Lower Bound (Skech) Some riple cro he cu and are e o 0 The maxerm of GEN are - cu (like a cu in - conneciviy). We add all riple which do no cro he cu. Le be he e of all cu.
66 Deerminiic Lower Bound (Skech) If M i a monoone wiching nework compuing GEN, we can naurally aociae wih each ae v a vecor which i 1 iff ha ae i reachable on a given cu.
67 Deerminiic Lower Bound (Skech) If M i a monoone wiching nework compuing GEN, we can naurally aociae wih each ae v a vecor which i 1 iff ha ae i reachable on a given cu.
68 Deerminiic Lower Bound (Skech) If M i a monoone wiching nework compuing GEN, we can naurally aociae wih each ae v a vecor which i 1 iff ha ae i reachable on a given cu. On here i a live pah o v o v
69 Deerminiic Lower Bound (Skech) If M i a monoone wiching nework compuing GEN, we can naurally aociae wih each ae v a vecor which i 1 iff ha ae i reachable on a given cu. On here i no live pah o v o v
70 Deerminiic Lower Bound (Skech) The vecor are called reachabiliy vecor. They ell u exacly how he nework operae on he cu. To ge a lower bound, we fix a large e of YES inance of GEN: pyramid inance of heigh h. Le be he e of heigh-h pyramid.
71 Deerminiic Lower Bound (Skech) we define anoher vecor.. 1. For every monoone wiching nework M compuing GEN and for each P in here i a ae.. 2. depend only on coordinae in P, and ha zero correlaion wih any funcion depending only on h coordinae 3.
72 Deerminiic Lower Bound (Skech) 1. For every monoone wiching nework M compuing GEN and for each P in here i a ae.. he ae mu perform a lo of work eparaing he pyramid P from all of he cu (I i highly pecific o P)
73 Deerminiic Lower Bound (Skech) 2. depend only on coordinae in P, and ha zero correlaion wih any funcion depending only on h coordinae. for any wo diinc pyramid ha hare a mo h coordinae, he funcion and are uncorrelaed (or orhogonal)
74 Summing hi inequaliy over all ae v and uing 1 give he lower bound. Deerminiic Lower Bound (Skech) 3. (uing orhogonaliy from 2) any ae can only be pecific (in he ene of 1) o a mo pyramid
75 Deerminiic Lower Bound (Skech) How do we come up wih hee funcion? For a fixed pyramid P, we inroduce new funcion for each riple l in P which aify a propery we call nicene, along wih wo oher properie: 1. For any,, and have he ame Fourier coefficien up o level h. 2. i mall. Nicene: If C i a cu and l croe C, hen
76 Average Cae Compuaion Le f be a boolean funcion and M a wiching nework. Le D be a diribuion on M compue f wih error on D if
77 Our Diribuion on GEN Inpu The diribuion we ue i uppored on a e of graph inance, and a e of maxerm. Definiion: Fix a DAG G wih fan-in 0 or 2. Define he diribuion a follow: wih probabiliy ½ chooe eiher a random cu inance or a random graph inance of GEN iomorphic o G
78 Our Diribuion on GEN Inpu The diribuion we ue i uppored on a e of graph inance, and a e of maxerm. Definiion: Fix a DAG G wih fan-in 0 or 2. Define he diribuion a follow: wih probabiliy ½ chooe eiher a random cu inance or a random graph inance of GEN iomorphic o G We vary he complexiy of he GEN problem by chooing graph wih differen reverible pebbling number.
79 Diribuion (Picure!) [N]
80 Random Cu [N]
81 Random Cu All riple added excep hoe ha cro he cu [N]
82 Random G-Inance [N]
83 Random G-Inance [N]
84 Random G-Inance Graph ha m < N verice Many poible iomorphic inance on [N] [N]
85 Lower bounding he average cae If he circui i allowed o error, here are wo baic difficulie in he previou proof: 1. The circui migh no accep all pyramid (no longer complee) 2. The circui migh no rejec all cu (no longer ound)
86 Lower bounding he average cae If he circui i allowed o error, here are wo baic difficulie in he previou proof: 1. The circui migh no accep all pyramid (no longer complee) 2. The circui migh no rejec all cu (no longer ound) The fir problem i eay o handle --- we can ill find a large ube of pyramid ha i acceped by he circui, on average.
87 Lower bounding he average cae If he circui i allowed o error, here are wo baic difficulie in he previou proof: 1. The circui migh no accep all pyramid (no longer complee) 2. The circui migh no rejec all cu (no longer ound) The econd problem i much harder, and overcoming i i our main echnical conribuion.
88 Lower bounding he average cae If he circui i allowed o error, here are wo baic difficulie in he previou proof: 1. The circui migh no accep all pyramid (no longer complee) 2. The circui migh no rejec all cu (no longer ound) The econd problem i much harder, and overcoming i i our main echnical conribuion. The problem arie in a echnical par of he conrucion of he vecor, and fixing i i ricky.
89 Puing i All Togeher: Main Theorem Main Theorem: Le N, m, h be poiive ineger and, non-negaive real parameer aifying Le G be a DAG wih fan-in 0 or 2, m verice, and reverible pebbling number h. Any monoone wiching nework compuing GEN on [N] wih error on ha a lea ae.
90 Corollarie All corollarie are obained by chooing graph wih differen pebbling number and eing parameer appropriaely.
91 Corollarie All corollarie are obained by chooing graph wih differen pebbling number and eing parameer appropriaely. Pyramid of heigh
92 Corollarie All corollarie are obained by chooing graph wih differen pebbling number and eing parameer appropriaely. Pyramid of heigh Pyramid of heigh h = N/2
93 Corollarie All corollarie are obained by chooing graph wih differen pebbling number and eing parameer appropriaely. Pyramid of heigh Pyramid of heigh h = N/2 Direced pah of lengh N
94 Corollarie All corollarie are obained by chooing graph wih differen pebbling number and eing parameer appropriaely. Pyramid of heigh Pyramid of heigh h = N/2 Direced pah of lengh N All lower bound are over wih error
95 Furher Direcion Framework preened only applie o GEN -- can we exend i o oher funcion?
96 Furher Direcion Framework preened only applie o GEN -- can we exend i o oher funcion? Can we opimize he lower bound furher? (In paricular, we loe a quare roo due o Cauchy-Schwarz)
97 Furher Direcion Framework preened only applie o GEN -- can we exend i o oher funcion? Can we opimize he lower bound furher? (In paricular, we loe a quare roo due o Cauchy-Schwarz) Doe hi proof have a more direc (read: combinaorial) analog in he world of monoone circui?
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