Something for almost nothing: Advances in sublinear time algorithms Ronitt Rubinfeld MIT and Tel Aviv U. Algorithms for REALLY big data No time What can we hope to do without viewing most of the data? Small world phenomenon • The social network is a graph: – “node’’ is a person – “edge’’ between people that know each other • “6 degrees of separation’’ • Are all pairs of people connected by path of distance at most 6? Vast data • Impossible to access all of it • Accessible data is too enormous to be viewed by a single individual • Once accessed, data can change The Gold Standard • Linear time algorithms: – for inputs encoded by n bits/words, allow cn time steps (constant c) • Inadequate… What can we hope to do without viewing most of the data? • Can’t answer “for all” or “exactly” type statements: • are all individuals connected by at most 6 degrees of separation? • exactly how many individuals on earth are left-handed? • Compromise? • is there a large group of individuals connected by at most 6 degrees of separation? • approximately how many individuals on earth are left-handed? What types of approximation? Property Testing • Does the input object have crucial properties? • Example Properties: • • • • • • Clusterability, Small diameter graph, Close to a codeword, Linear or low degree polynomial function, Increasing order Lots and lots more… “In the ballpark” vs. “out of the ballpark” tests • Property testing: Distinguish inputs that have specific property from those that are far from having that property • Benefits: – Can often answer such questions much faster – May be the natural question to ask • When some “noise” always present • When data constantly changing • Gives fast sanity check to rule out very “bad” inputs (i.e., restaurant bills) • Model selection problem in machine learning Examples • Can test if a function is a homomorphism in CONSTANT TIME [Blum Luby R.] • Can test if the social network has 6 degrees of separation in CONSTANT TIME [Parnas Ron] Constructing a property tester: • Find characterization of property that is • Efficiently (locally) testable • Robust • objects that have the property satisfy characterization, • and objects far from having the property are unlikely to PASS Usually the bigger challenge Example: Homomorphism property of functions • A “bad” testing characterization: ∀x,y f(x)+f(y) = f(x+y) • Another bad characterization: For most x f(x)+f(1) = f(x+1) • Good characterization: For most x,y f(x)+f(y) = f(x+y) Example: 6 degrees of separation • A “bad” testing characterization: For every node, all other nodes within distance 6. • Another bad one: For most nodes, all other nodes within distance 6. • Good characterization: For most nodes, there are many other nodes within distance 6. An example in depth Monotonicity of a sequence • Given: list y1 y2 ... yn • Question: is the list sorted? • Clearly requires n steps – must look at each yi Monotonicity of a sequence • Given: list y1 y2 ... yn • Question: can we quickly test if the list close to sorted? What do we mean by ``quick’’? • query complexity measured in terms of list size n • Our goal (if possible): • Very small compared to n, will go for clog n What do we mean by “close’’? Definition: a list of size n is .99-close to sorted if can delete at most .01n values to make it sorted. Otherwise, .99-far. Sorted: 1 2 4 5 7 11 14 19 20 21 23 38 39 45 Close: 1 4 2 5 7 11 14 19 20 39 23 21 38 45 1 4 5 7 11 14 19 20 23 38 45 Far: 45 39 23 1 38 4 5 21 20 19 2 7 11 14 1 4 5 7 11 14 Requirements for algorithm: • pass sorted lists • if list passes test, can change at most .01 fraction of list to make it sorted An attempt: • Proposed algorithm: • Pick random i and test that yi≤yi+1 • Bad input type: • 1,2,3,4,5,…j, 1,2,3,4,5,….j, 1,2,3,4,5,…j, 1,2,3,4,5,…,j • Difficult for this algorithm to find “breakpoint” • But other algorithms work well yi i A second attempt: • Proposed algorithm: • Pick random i<j and test that yi≤yj • Bad input type: • n/m groups of m elements m,m-1,m-2,…,1, 2m,2m-1,2m-2,…m+1, 3m,3m-1,3m-2,…, • must pick i,j in same group • need at least (n/m)1/2 choices to do this yi i A test that works • The test: (for distinct yi) • Test several times: • • • • • Pick random i Look at value of yi Do binary search for yi Does the binary search find any inconsistencies? If yes, FAIL Do we end up at location i? If not FAIL • Pass if never failed • Running time: O(log n) time • Why does this work? • If list is in order, then test always passes • If the test passes on choice i and j, then yi and yj are in correct order • Since test usually passes, most yi’s in the right order Many more properties studied! • Graphs, functions, point sets, strings, … • Amazing characterizations of problems testable in graph and function testing models! Properties of functions • Linearity and low total degree polynomials [Blum Luby R.] [Bellare Coppersmith Hastad Kiwi Sudan] [R. Sudan] [Arora Safra] [Arora Lund Motwani Sudan Szegedy] [Arora Sudan] ... • Functions definable by functional equations – trigonometric, elliptic functions [R.] • All locally characterized affine invariant function classes! [Bhattacharyya Fischer Hatami Hatami Lovett] • Groups, Fields [Ergun Kannan Kumar R. Viswanathan] • Monotonicity [EKKRV] [Goldreich Goldwasser Lehman Ron Samorodnitsky] [Dodis Goldreich Lehman Ron Raskhodnikova Samorodnitsky][Fischer Lehman Newman Raskhodnikova R. Samorodnitsky] [Chakrabarty Seshadri]… • Convexity, submodularity [Parnas Ron R.] [Fattal Ron] [Seshadri Vondrak]… • Low complexity functions, Juntas [Parnas Ron Samorodnitsky] [Fischer Kindler Ron Safra Samorodnitsky] [Diakonikolas Lee Matulef Onak R. Servedio Wan]… Properties of sparse and general graphs • Easily testable dense and hyperfinite graph properties are completely characterized! [Alon Fischer Newman Shapira][Newman Sohler] • General Sparse graphs: bipartiteness, connectivity, diameter, colorability, rapid mixing, triangle free,… [Goldreich Ron] [Parnas Ron] [Batu Fortnow R. Smith White] [Kaufman Krivelevich Ron] [Alon Kaufman Krivelevich Ron]… • Tools: Szemeredi regularity lemma, random walks, local search, simulate greedy, borrow from parallel algorithms Some other combinatorial properties • Set properties – equality, distinctness,... • String properties – edit distance, compressibility,… • Metric properties – metrics, clustering, convex hulls, embeddability... • Membership in low complexity languages – regular languages, constant width branching programs, context-free languages, regular tree languages… • Codes – BCH and dual-BCH codes, Generalized ReedMuller,… “Traditional” approximation • Output number close to value of the optimal solution (not enough time to construct a solution) • Some examples: • • • • • Minimum spanning tree, vertex cover, max cut, positive linear program, edit distance, … Example: Vertex Cover • Given graph G(V,E), a vertex cover (VC) C is a subset of V such that it “touches” every edge. • What is minimum size of a vertex cover? • NP-complete • Poly time multiplicative 2-approximation based on relationship of VC and maximal matching Vertex Cover and Maximal Matching • Maximal Matching: • M ⊆ E is a matching if no node in in more than one edge. • M is a maximal matching if adding any edge violates the matching property • Note: nodes matched by M form a pretty good Vertex Cover! • Vertex cover must include at least one node in each edge of M • Union of nodes of M form a vertex cover • So |M| ≤ VC ≤ 2 |M| “Classical” approximation examples • Can get CONSTANT TIME approximation for vertex cover on sparse graphs! • Output y which is at most 2∙ OPT + ϵn How? • Oracle reduction framework [Parnas Ron] • • Construct “oracle” that tells you if node u in 2-approx vertex cover Use oracle + standard sampling to estimate size of cover But how do you implement the oracle? Implementing the oracle – two approaches: • Sequentially simulate computations of a local distributed algorithm [Parnas Ron] • Figure out what greedy maximal matching algorithm would do on u [Nguyen Onak] Greedy algorithm for maximal matching • Algorithm: • ←∅ • For every edge (u,v) • If neither of u or v matched • Add (u,v) to M • Output M • Why is M maximal? • If e not in M then either u or v already matched by earlier edge Implementing the Oracle via Greedy • To decide if edge e in matching: • Must know if adjacent edges that come before e in the ordering are in the matching • Do not need to know anything about edges coming after • Arbitrary edge order can have long dependency chains! 1 2 4 8 25 36 47 88 89 Odd or even steps from beginning? 110 111 112 113 Breaking long dependency chains [Nguyen Onak] • Assign random ordering to edges • Greedy works under any ordering • Important fact: random order has short dependency chains Better Complexity for VC • Always recurse on least ranked edge first • Heuristic suggested by [Nguyen Onak] • Yields time poly in degree [Yoshida Yamamoto Ito] • Additional ideas yield query complexity nearly linear in average degree for general graphs [Onak Ron Rosen R.] Further work • More complicated arguments for maximum matching, set cover, positive LP… [Parnas Ron + Kuhn Moscibroda Wattenhofer] [Nguyen Onak] [Yoshida Yamamoto Ito] Can dependence be made poly in average degree? • Even better results for hyperfinite graphs [Hassidim Kelner Nguyen Onak][Newman Sohler] • e.g., planar No samples What if data only accessible via random samples? Distributions Play the lottery? Is the lottery unfair? • From Hitlotto.com: Lottery experts agree, past number histories can be the key to predicting future winners. True Story! • Polish lottery Multilotek • Choose “uniformly” at random distinct 20 numbers out of 1 to 80. • Initial machine biased • e.g., probability of 50-59 too small • Past results: http://serwis.lotto.pl:8080/archiwum/wyniki_wszystkie.php?id_gra=2 Thanks to Krzysztof Onak (pointer) and Eric Price (graph) New Jersey Pick 3,4 Lottery • New Jersey Pick k ( =3,4) Lottery. • Pick k digits in order. • 10k possible values. • Assume lottery draws iid • Data: • Pick 3 - 8522 results from 5/22/75 to 10/15/00 • 2-test gives 42% confidence • Pick 4 - 6544 results from 9/1/77 to 10/15/00. • fewer results than possible values • 2-test gives no confidence Distributions on BIG domains • Given samples of a distribution, need to know, e.g., • • • • entropy number of distinct elements “shape” (monotone, bimodal,…) closeness to uniform, Gaussian, Zipfian… • No assumptions on shape of distribution • i.e., smoothness, monotonicity, Normal distribution,… • Considered in statistics, information theory, machine learning, databases, algorithms, physics, biology,… Key Question • How many samples do you need in terms of domain size? • Do you need to estimate the probabilities of each domain item? • Can sample complexity be sublinear in size of the domain? Rules out standard statistical techniques, learning distribution Our Aim: Algorithms with sublinear sample complexity Similarities of distributions Are p and q close or far? • p is given via samples • q is either • known to the tester (e.g. uniform) • given via samples Is p uniform? • Theorem: ([Goldreich Ron] [Batu p Fortnow R. Smith White] [Paninski]) samples Test Pass/Fail? Sample complexity of distinguishing = /2 1 from | − |1 > is ( ) • Nearly same complexity to test if p is any known distribution [Batu Fischer Fortnow Kumar R. 1 White]:“Testing identity” − 1 = Σ − Upper bound for L2 distance [Goldreich Ron] • L2 distance: − 2 2 = ∑ − • ||p-U||22 = S(pi -1/n)2 = Spi2 - 2Spi /n + S1/n2 = Spi2 - 1/n 2 ======== • Estimate collision probability to estimate L2 distance from uniform Testing uniformity [GR, Batu et. al.] • Upper bound: Estimate collision probability and use known relation between between L1 and L2 norms • Issues: • Collision probability of uniform is 1/n • Use O(sqrt(n)) samples via recycling • Comment: [P] uses different estimator • Easy lower bound: (n½) • Can get (n½/2) [P] Back to the lottery… plenty of samples! Is p uniform? • Theorem: ([Goldreich Ron][Batu p Fortnow R. Smith White] [Paninski]) samples Test Pass/Fail? Sample complexity of distinguishing p=U from |p-U|1> is (n1/2) • Nearly same complexity to test if p is any known distribution [Batu Fischer Fortnow Kumar R. White]: “Testing identity” Testing closeness of two distributions: Transactions of 20-30 yr olds trend change? Transactions of 30-40 yr olds Testing closeness p q Test Pass/Fail? Theorem: ([BFRSW] [P. Valiant]) Sample complexity of distinguishing p=q from ||p-q||1 > ~ 2/3 is (n ) Why so different? • Collision statistics are all that matter • Collisions on “heavy” elements can hide collision statistics of rest of the domain • Construct pairs of distributions where heavy elements are identical, but “light” elements are either identical or very different Additively estimate distance? Output ||p-q||1 ± need (n/log n) samples [G. Valiant P. Valiant] Information theoretic quantities • Entropy • Support size Information in neural spike trails [Strong, Koberle, de Ruyter van Steveninck, Bialek ’98] • Each application of stimuli gives sample of signal (spike trail) Neural signals time • Entropy of (discretized) signal indicates which neurons respond to stimuli Compressibility of data Can we get multiplicative approximations? • In general, no…. • 0 entropy distributions are hard to distinguish • What if entropy is bigger? • Can g-multiplicatively approximate the entropy with Õ(n1/g2) samples (when entropy >2g/) [Batu Dasgupta R. Kumar] • requires (n1/g2) [Valiant] • better bounds when support size is small [Brautbar Samorodnitsky] • Similar bounds for estimating support size [Raskhodikova Ron R. Smith] [Raskhodnikova Ron Shpilka Smith] More properties: • Independence: [Batu Fischer Fortnow Kumar R. White] • Limited Independence: [Alon Andoni Kaufman Matulef R. Xie] [Haviv Langberg] • K-flat distributions [Levi Indyk R.] • K-modal distributions [Daskalakis Diakonikolas Servedio] • Poisson Binomial Distributions [Daskalakis Diakonikolas Servedio] • Monotonicity over general posets [Batu Kumar R.] [Bhattacharyya Fischer R. P. Valiant] • Properties of multiple distributions [Levi Ron R.] Conclusion: • For many problems, we need a lot less time and samples than one might think! • Many cool ideas and techniques have been developed • Lots more to do! Thank you!