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or decrease key fibonacci and improvements that achieve o 1 time pairing heaps that achieve o 2 2 sqrt loglog n time elmasry sort heaps that achieve o loglog n time all decrease key operations of these trees pair small trees into bigger trees what distingushes them is how they do this pairing heaps repeatedly pair trees recursively repeat until one tree remains elmasry sort heaps break roots into blocks of size log n chain all roots of a block together for fibonacci heaps pair heaps whose roots have the same number of children note that the fibonacci pair based on tree size not key value and use an augmentation of loglog n bits in each root seems like you augmented bits fast fredman considered this in 1999 proving o 1 decrease key implies omega loglog n bits for some classes of heaps namely pairing heaps but lower bound requires assumption that every time a pair of roots have their values compared their heaps are combined this contribution of iacono and özkan is a new lower bound without such the requirement that trees are combined when compared but with a new restriction must pay pointer model costs this result yields lower bound of omega loglog n for pairing and sort heaps but says nothing about fibonacci heaps because they cheat they do not pay pointer model costs a pointer model version would build a tree above the array which would have height loglog n and give decrease key a similar running time the actual proof is adversarial and considers a whole family of distinct fibonacci heaps that must grow larger than the total number of possible heaps a contradiction posted by luca aceto at 8 08 pm no comments email this blogthis share to x share to facebook share to pinterest icalp day 2 guest post by andrew winslow andrew winslow was one of the colleagues who kindly answered my call for guest bloggers from icalp 2014 here is guest post on the second conference day enjoy it invited talk victor kuncak victor gave a talk entitled verifying and synthesizing software with recursive functions about work with his students including ruzica piskac phillipe suter eva darulova ettienne kneuss and andrej spielmann the work focused on a number of problems in the area of turning a specification into a program that meets it he did a great job of breaking out four types of problems they re working on run time compile time have specification c and program p assertion checking check p meets c prove correctness prove p always meets c have specification c constraint programming carry out execution according to c program synthesis create p meeting c their work is done in scala and implemented in the leon system out of the four types of problems victor indicated that the first two first row of the table are easier of course easier doesn t mean easy and he briefly covered a few of the ideas induction arithmetic algebraic reasoning sat solvers integer linear programs used to achieve good empirical performance a few seconds to prove correctness programs of 100 lines of code or more and counterexamples in the case of incorrect code constraint programming was next and he gave the example of programming red black tress via constraints using as constraints the five properties a red black tree must have insertion for instance would be encoded as a combination of the five properties plus the fact that the new tree must have the elements of the old tree plus the inserted item although the performance was not great he stressed that the applications of constraint programming are more along the lines of prototyping new data structures test case generation etc one cool trick was using the fact that a solution is a counterexample of the negation of the constraints reducing the problem to one in program verification finally he covered most extreme problem writing code based on a specification here he had some really nice examples generating code for converting dates and times simple data structures like sorted lists etc in just a few seconds many ideas were used but i was especially impressed by his mention of their work on floating point representations synthesizing programs that avoided rounding errors andreas björklund and thore husfeldt shortest two disjoint paths in polynomial time andreas gave the talk for this best paper award winner in track a the problem considered is a very simple one given an undirected graph and two pairs of vertices s1 t1 and s2 t2 find a pair of disjoint paths whose total length is minimized several problems near to this one are np hard e g minimizing the maximum of the two path lengths so placing the problem into p is the major thrust of the paper the algorithms given run in o n 11 and o n 22 time for vertex and edge disjoint paths respectively the approach of the algorithm is almost entirely matrix based focusing on solving a cycle cover problem on the input graph with self loops added to each vertex except that doesn t quite work andreas did a great job of leading us on a journey where obstacle after obstacle to the approach was eliminated many of the issues relate to the complexity of computing permanent indeed one of the major technical contributions is proving that computing the permanent of a matrix over a certain type of ring is in p in the end the necessary permanent computation is replaced with a polynomial number of determinant computations giving the large but polynomial running time in perhaps the most natural course of events someone asked after the talk whether the technique might work for the shortest three disjoint paths so natural was the question that a followup slide for the question was already prepared the conclusion was probably not but andreas posed a first problem to try in this direction given an undireced graph compute the parity of the number of cycle covers having c cycles with c divisible by 4 mitsuru kusumoto and yuichi yoshida testing forest isomorphism in the adjacency list model mitsuru started by introducing the notion of property testing on graphs computing whether the graph has a given property answering correctly at least 2 3rds of the time while using only a small number of queries on the graph in this setting even the representation of a graph as an adjacency matrix or list matters as checking whether a given edge is present takes too long with an adjacent list repesentation kusumoto and yoshida considered the problem of testing whether two forests are isomorphic it was known that in general graphs this problem requires omega sqrt n queries to the graphs and only o 1 if the graphs have bounded degree for forests they achieve an algorithm that uses only o polylog n queries and prove omega sqrt log n are needed the algorithm works by first performing a special type of partitioning of the input graph then testing whether each subgraph is isomorphic the partitioning is done in a way that ensures that if the two input graphs are significantly far away from being isomorphic then some pair of their subgraphs in their partitions are too rom aschner matthew j katz bounded angle spanning tree modeling networks with angular constraints rom gave a nicely visual talk about a problem in wireless network design in the plane the input is a set of points and the output is a euclidean minimum spanning tree mst with an additional constraint the smallest angle spanning all of the edges is at most some fixed constant b a b mst another way to think about this constraint is that the largest angle between any pair of adjacent incoming edges must be at least 2 π b for b π 3 a b mst may not even exist three points at the vertices of an equilateral triangle require at least one of the points to have two incoming edges with interior angle π 3 for b 8 π 5 the b mst is simply the minimum spanning tree as the upper degree bound of 5 on every node in an mst ensures the smallest angle spanning all of the edges is at least 2 π 2 π 5 8 π 5 finally for b 1 2 there exists a simple family of examples n 1 points placed collinearly ε distance apart and a final point distance 1 from the shared line that causes the b mst to have total weight ε n 1 1 and b mst to have total weight 1 n 1 n 1 so the non trivial b are those in the interval 1 2 8 5 pi for these aschner and katz prove both np hardness and approximation results they prove that finding the b mst for b π and b 2 π 3 are both np hard using a reduction from hamiltonian path in square and hexagonal grid graphs respectively they also achieve constant factor approximation algorithms for b π 2 π 3 π 2 the approximation algorithms use a common sequence of steps compute the mst turn the mst into a hamiltonian cycle via a tour around the boundary of the mst decompose the points into consecutive groups along the cycle for each group compute a set of edges connecting the points and minimizing total edge length connect adjacent groups using the shortest edge possible step 4 requires a careful selection of group sizes as larger groups give greater potential for shorter collections of edges but for large enough b actually connecting the points may not be possible the resulting approximation ratios are 16 and lower and can further be reduced by a neat pigeonhole argument regarding shofting the groupings along the tour canal tour and conference dinner the day concluded with a tour through the canals of copenhagen and dinner at the odd fellow palace the 2014 gödel prize was awarded to ronald fagin amnon lotem and moni naor moni gave a great historical and technical talk on the work for which they received the award including the threshold algorithm ronald fagin also spoke briefly about michael franklin the missing fourth author of the work and his role in facilitating the successful collaboration posted by luca aceto at 6 36 am no comments email this blogthis share to x share to facebook share to pinterest wednesday july 09 2014 july 8 2014 icalp review guest post by clément canonne here is a first guest post from icalp 2014 kindly written by clément canonne enjoy it this post is also available in pdf any typo or typesetting issue with the html version is my responsibility on dnf approximators for monotone boolean functions eric blais in this talk eric blais presented joint work with johan håstad rocco servedio and li yang tan concerned with boolean functions more precisely the simplest representations of functions dnf disjunctive normal form formulas for a little bit of background recall that a boolean function f 0 1 n 0 1 defined on the hypercube 2 is a dnf if it can be written as that is as an or of and s one can also see such functions as being eactly those taking value 1 on an union of subcubes if one prefers i will not argue with one a nice property of dnf formulas is that they are arguably amongst the simplest of all representations of boolean functions while formulas of depth 3 are a nightmare dnfs have been extensively studied and by now everything is known about them well almost everything indeed amidst other facts we have that theorem 1 folklore every boolean function can be computed by a dnf of size 2 n 1 theorem 2 lupanov 61 this is tight parity n needs that much theorem 3 korshunov 81 kuznetsov 83 a random boolean function can be computed by dnfs of size θ 2 n log n and requires that much so are we done yet the mere presence of eric in the auditorium was a clear hint that all was not settled and as it turns out if the picture is well understood for exact computation of boolean functions by dnfs what about approximate representation of a function that is what about the size required to approximate a boolean function by a dnf if one allows error ε as a fraction of the inputs this leads to the notion of dnf approximator complexity and here again some results much more recent results theorem 4 blais tan 13 every boolean function can be approximated by a dnf of size o 2 n log n furthermore our all friend parity n only needs dnf size o 2 1 2ε n that s way better than 2 n 1 so again are we done here and again not quite this brings us to the main point of the paper namely what about monotone functions can they be computed more efficiently approximated more efficiently recall that a boolean function f is monotone if x y implies f x f y where is the coordinate wise partial order on bit strings as a starter no theorem 5 folklore every monotone boolean function can be computed by a dnf of size o 2 n n 1 2 using subcubes rooted on each min term and again this is tight for parity furthermore and quite intuitively using negations does not buy you anything to compute a monotone function and why should it indeed theorem 6 quine 54 to compute monotone boolean functions monotone dnfs are the best amongst dnfs not surprising i suppose well it s a whole new game when one one again asks only for approximations and that s the gist of the paper presented here first of all drastic savings in the size of the formulas theorem 7 blais hastad servedio tan 14 every monotone boolean function can be approximated by a dnf of size o 2 n 2 ω n 1 2 eric gave a high level view of the proof again it works by considering the subcubes rooted on each min term but in two steps regularity lemma the world would be much simpler if all subcubes were rooted on the same level of the hypercube so first reduce it to this case writing f f 1 f k each f i has this property then approximate independently each f i using a probabilistic argument via random sampling to prove there exists a good approximator for all f i s and then stitching them together and they also show it is tight this time with the majority function maj n the proof goes by a counting argument and concentration of measure on the hypercube every or almost every input is on the middle belt of the hypercube but each subcube thus has to be rooted there and each cannot cover too much so many are needed so approximation does buy us a lot but clearly using negations shouldn t should it why would allowing non monotone dnf s to approximate monotone functions ever help hint it does yep theorem 8 blais hastad servedio tan 14 for every n there exists ε n and f 0 1 6n 0 1 such that f can be ε n approximated by dnfs of size o n any monotone dnf ε n approximating f must have size ω n 2 take that intuition the upshot exact computation and approximate computation have intrinsically very different properties eric then concluded with an open question namely how to improve better understand the gap between approximating functions with monotone dnf vs approximating them with general dnf s the currently known gap in the size being quite huge almost exponential additionally how to get a separation as in the mind boggling theorem above but changing the quantifiers that is for a constant ε independent of n also can someone fix my intuition i think it s broken 1 http www cs columbia edu rocco papers icalp14 html 2 not this one posted by luca aceto at 9 44 pm no comments email this blogthis share to x share to facebook share 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