HAL (Le Centre pour la Communication Scientifique Directe), Dec 1, 2009
The covering and boundedness problems for branching vector addition systems are shown complete fo... more The covering and boundedness problems for branching vector addition systems are shown complete for doubly-exponential time.
Scheduling a sequence of jobs released over time when the processing time of a job is only known ... more Scheduling a sequence of jobs released over time when the processing time of a job is only known at its completion is a classical problem in CPU scheduling in time sharing operating systems. A widely used measure for the responsiveness of the system is the average flow time of the jobs, that is, the average time spent by jobs in the system between release and completion. The Windows NT and the Unix operating system scheduling policies are based on the Multilevel Feedback algorithm. In this article, we prove that a randomized version of the Multilevel Feedback algorithm is competitive for single and parallel machine systems, in our opinion providing one theoretical validation of the goodness of an idea that has proven effective in practice along the last two decades. The randomized Multilevel Feedback algorithm (RMLF) was first proposed by Kalyanasundaram and Pruhs for a single machine achieving an O(log n log log n) competitive ratio to minimize the average flow time against the on-line adaptive adversary, where n is the number of jobs that are released. We present a version of RMLF working for any number m of parallel machines. We show for RMLF a first O(log n log n m) competitiveness result against the oblivious adversary on parallel machines. We also show that the same RMLF algorithm surprisingly achieves a tight O(log n) competitive ratio against the oblivious adversary on a single machine, therefore matching the lower bound for this case.
We consider testing directed graphs for being Eulerian in the orientation model introduced in [15... more We consider testing directed graphs for being Eulerian in the orientation model introduced in [15]. Despite the local nature of the property of being Eulerian, it turns out to be significantly harder for testing than other properties studied in the orientation model. We show a non-constant lower bound on the query complexity of 2-sided tests and a linear lower bound on the query complexity of 1-sided tests for this property. On the positive side, we give several 1-sided and 2-sided tests, including a sub-linear query complexity 2-sided test for general graphs. For special classes of graphs, including bounded-degree graphs and expander graphs, we provide improved results. In particular, we give a 2-sided test with constant query complexity for dense graphs, as well as for expander graphs with a constant expansion parameter.
In this paper we establish an exponential lower bound on the size of syntactic non-deterministic ... more In this paper we establish an exponential lower bound on the size of syntactic non-deterministic read d-times branching programs for d ≤ log n/10 5 computing a class of monotone CNFs with a linear number of clauses. This result provides the first separation of classes NP and co-NP for syntactic branching programs with a logarithmic repetition and the first separation of syntactic non-deterministic branching programs with a logarithmic repetition from small monotone CNFs.
Symposium on Theoretical Aspects of Computer Science, 2017
Distribution testing deals with what information can be deduced about an unknown distribution ove... more Distribution testing deals with what information can be deduced about an unknown distribution over {1,. .. , n}, where the algorithm is only allowed to obtain a relatively small number of independent samples from the distribution. In the extended conditional sampling model, the algorithm is also allowed to obtain samples from the restriction of the original distribution on subsets of {1,. .. , n}. In 2015, Canonne, Diakonikolas, Gouleakis and Rubinfeld unified several previous results, and showed that for any property of distributions satisfying a "decomposability" criterion, there exists an algorithm (in the basic model) that can distinguish with high probability distributions satisfying the property from distributions that are far from it in variation distance. We present here a more efficient yet simpler algorithm for the basic model, as well as very efficient algorithms for the conditional model, which until now was not investigated under the umbrella of decomposable properties. Additionally, we provide an algorithm for the conditional model that handles a much larger class of properties. Our core mechanism is a way of efficiently producing an interval-partition of {1,. .. , n} that satisfies a "fine-grain" quality. We show that with such a partition at hand we can directly move forward with testing individual intervals, instead of first searching for the "correct" partition of {1,. .. , n}.
Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices ar... more Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst case guarantees. We propose a new data structure, the dynamic tree (D-tree), together with algorithms to construct and maintain it. The D-tree is the first data structure that scales to fully dynamic graphs with millions of vertices and edges and, on average, answers connectivity queries much faster than data structures with worst case guarantees.
A string α ∈ Σ n is called p-periodic, if for every i, j ∈ {1,. .. , n}, such that i ≡ j mod p, α... more A string α ∈ Σ n is called p-periodic, if for every i, j ∈ {1,. .. , n}, such that i ≡ j mod p, αi = αj, where αi is the i-th place of α. A string α ∈ Σ n is said to be period(≤ g), if there exists p ∈ {1,. .. , g} such that α is p-periodic. An property tester for period(≤ g) is a randomized algorithm, that for an input α distinguishes between the case that α is in period(≤ g) and the case that one needs to change at least-fraction of the letters of α, so that it will become period(≤ g). The complexity of the tester is the number of letter-queries it makes to the input. We study here the complexity of testers for period(≤ g) when g varies in the range 1,. .. , n 2. We show that there exists a surprising exponential phase transition in the query complexity around g = log n. That is, for every δ > 0 and for each g, such that g ≥ (log n) 1+δ , the number of queries required and sufficient for testing period(≤ g) is polynomial in g. On the other hand, for each g ≤ logn 4 , the number of queries required and sufficient for testing period(≤ g) is only poly-logarithmic in g. We also prove an exact asymptotic bound for testing general periodicity. Namely, that 1-sided error, non adaptive-testing of periodicity (period(≤ n 2)) is Θ(√ n log n) queries.
We consider testing directed graphs for being Eulerian in the orientation model introduced in [15... more We consider testing directed graphs for being Eulerian in the orientation model introduced in [15]. Despite the local nature of the property of being Eulerian, it turns out to be significantly harder for testing than other properties studied in the orientation model. We show a non-constant lower bound on the query complexity of 2-sided tests and a linear lower bound on the query complexity of 1-sided tests for this property. On the positive side, we give several 1-sided and 2-sided tests, including a sub-linear query complexity 2-sided test for general graphs. For special classes of graphs, including bounded-degree graphs and expander graphs, we provide improved results. In particular, we give a 2-sided test with constant query complexity for dense graphs, as well as for expander graphs with a constant expansion parameter.
We prove almost tight bounds on the length of paths in 2-edge-connected cubic graphs. Concretely,... more We prove almost tight bounds on the length of paths in 2-edge-connected cubic graphs. Concretely, we show that (i) every 2-edge-connected cubic graph of size n has a path of length Ω log 2 n log log n , and (ii) there exists a 2-edge-connected cubic graph, such that every path in the graph has length O(log 2 n)
Journal of Computer and System Sciences, Feb 1, 2013
The covering and boundedness problems for branching vector addition systems are shown complete fo... more The covering and boundedness problems for branching vector addition systems are shown complete for doubly-exponential time.
We study the query complexity of testing for properties defined by read once formulas, as instanc... more We study the query complexity of testing for properties defined by read once formulas, as instances of massively parametrized properties, and prove several testability and non-testability results. First we prove the testability of any property accepted by a Boolean read-once formula involving any bounded arity gates, with a number of queries exponential in ǫ, doubly exponential in the arity, and independent of all other parameters. When the gates are limited to being monotone, we prove that there is an estimation algorithm, that outputs an approximation of the distance of the input from satisfying the property. For formulas only involving And/Or gates, we provide a more efficient test whose query complexity is only quasipolynomial in ǫ. On the other hand, we show that such testability results do not hold in general for formulas over non-Boolean alphabets; specifically we construct a property defined by a read-once arity 2 (non-Boolean) formula over an alphabet of size 4, such that any 1/4-test for it requires a number of queries depending on the formula size. We also present such a formula over an alphabet of size 5 that additionally satisfies a strong monotonicity condition.
Distribution testing deals with what information can be deduced about an unknown distribution ove... more Distribution testing deals with what information can be deduced about an unknown distribution over {1,. .. , n}, where the algorithm is only allowed to obtain a relatively small number of independent samples from the distribution. In the extended conditional sampling model, the algorithm is also allowed to obtain samples from the restriction of the original distribution on subsets of {1,. .. , n}. In 2015, Canonne, Diakonikolas, Gouleakis and Rubinfeld unified several previous results, and showed that for any property of distributions satisfying a "decomposability" criterion, there exists an algorithm (in the basic model) that can distinguish with high probability distributions satisfying the property from distributions that are far from it in the variation distance. We present here a more efficient yet simpler algorithm for the basic model, as well as very efficient algorithms for the conditional model, which until now was not investigated under the umbrella of decomposable properties. Additionally, we provide an algorithm for the conditional model that handles a much larger class of properties. Our core mechanism is a way of efficiently producing an interval-partition of {1,. .. , n} that satisfies a "fine-grain" quality. We show that with such a partition at hand we can directly move forward with testing individual intervals, instead of first searching for the "correct" partition of {1,. .. , n}.
Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices ar... more Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst case guarantees. We propose a new data structure, the dynamic tree (D-tree), together with algorithms to construct and maintain it. The D-tree is the first data structure that scales to fully dynamic graphs with millions of vertices and edges and, on average, answers connectivity queries much faster than data structures with worst case guarantees.
Society for Industrial and Applied Mathematics eBooks, 2020
Please refer to published version for the most recent bibliographic citation information. If a pu... more Please refer to published version for the most recent bibliographic citation information. If a published version is known of, the repository item page linked to above, will contain details on accessing it.
1 1-sided which doesn’t work Let w ∈ {0, 1}n, i ∈ [n] and Q ⊆ [n]. We use wi to denote the i’th l... more 1 1-sided which doesn’t work Let w ∈ {0, 1}n, i ∈ [n] and Q ⊆ [n]. We use wi to denote the i’th letter of w and wQ to be the a string v ∈ {0, 1}|Q| such that, for every j ∈ [|Q|], vj = wk(j), where k(j) is the j’th smallest member of Q. Definition 1.1 (constraints, etc). A q-constraint is a pair C = (Q,U) where Q is a subset of [n] of size at most q, and U is a subset of {0, 1}|Q|. A word w ∈ {0, 1}n is said to violate C if wQ ∈ U . A q-formula F is a set of q-constraints, all of whose corresponding Q sets are distinct. The property PF is defined as the set of words that violate no member of F . We say that F is solvable if PF 6= ∅. Given a property P and a a set Q ⊆ [n], the natural constraint is CQ = (Q,UQ), where UQ ⊆ {0, 1}q is the set of strings v for which there exist no w ∈ P with wQ = v. A witness against a word w 6∈ P is a set Q so that w does not satisfy the natural constraint (Q,UQ). Similarly, given a set of subsets of [n], the corresponding natural formula is the set of...
A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T cont... more A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T controls another vertex y if there exists a directed path of length at most two from x to y. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu [8] in investigating the query complexity of finding a king, that is, the number of arcs in T one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least Ω(n 4/3) arcs and provided a strategy that queries at most O(n 3/2). While this upper bound has not yet been improved for the original problem, Biswas et al. [3] proved that with O(n 4/3) queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using O(n 4/3 polylog n) queries, we can identify a (1 2 + 2 17)-king. To achieve this goal we use a novel structural result for tournaments.
A string α ∈ Σ is called p-periodic, if for every i, j ∈ {1, . . . , n}, such that i ≡ j mod p, α... more A string α ∈ Σ is called p-periodic, if for every i, j ∈ {1, . . . , n}, such that i ≡ j mod p, αi = αj , where αi is the i-th place of α. A string α ∈ Σ is said to be period(≤ g), if there exists p ∈ {1, . . . , g} such that α is p-periodic. An property tester for period(≤ g) is a randomized algorithm, that for an input α distinguishes between the case that α is in period(≤ g) and the case that one needs to change at least -fraction of the letters of α, so that it will become period(≤ g). The complexity of the tester is the number of letter-queries it makes to the input. We study here the complexity of testers for period(≤ g) when g varies in the range 1, . . . , n 2 . We show that there exists a surprising exponential phase transition in the query complexity around g = log n. That is, for every δ > 0 and for each g, such that g ≥ (log n), the number of queries required and sufficient for testing period(≤ g) is polynomial in g. On the other hand, for each g ≤ logn 4 , the number...
We study the following question, communicated to us by Miklós Ajtai: Can all explicit (e.g., poly... more We study the following question, communicated to us by Miklós Ajtai: Can all explicit (e.g., polynomial time computable) functions f : ({0, 1} w ) 3 → {0, 1} w be computed by word circuits of constant size? Here, a word circuit is an acyclic circuit where each wire holds a word (i.e., an element of {0, 1} w ) and each gate G computes some binary operation gG : ({0, 1} w ) 2 → {0, 1} w , defined for all word lengths w. As our main result, we present an explicit function so that its w'th slice for any w ≥ 8 cannot be computed by word circuits with at most 4 gates. Also, we formally relate Ajtai's question to open problems concerning ACC 0 circuits.
HAL (Le Centre pour la Communication Scientifique Directe), Dec 1, 2009
The covering and boundedness problems for branching vector addition systems are shown complete fo... more The covering and boundedness problems for branching vector addition systems are shown complete for doubly-exponential time.
Scheduling a sequence of jobs released over time when the processing time of a job is only known ... more Scheduling a sequence of jobs released over time when the processing time of a job is only known at its completion is a classical problem in CPU scheduling in time sharing operating systems. A widely used measure for the responsiveness of the system is the average flow time of the jobs, that is, the average time spent by jobs in the system between release and completion. The Windows NT and the Unix operating system scheduling policies are based on the Multilevel Feedback algorithm. In this article, we prove that a randomized version of the Multilevel Feedback algorithm is competitive for single and parallel machine systems, in our opinion providing one theoretical validation of the goodness of an idea that has proven effective in practice along the last two decades. The randomized Multilevel Feedback algorithm (RMLF) was first proposed by Kalyanasundaram and Pruhs for a single machine achieving an O(log n log log n) competitive ratio to minimize the average flow time against the on-line adaptive adversary, where n is the number of jobs that are released. We present a version of RMLF working for any number m of parallel machines. We show for RMLF a first O(log n log n m) competitiveness result against the oblivious adversary on parallel machines. We also show that the same RMLF algorithm surprisingly achieves a tight O(log n) competitive ratio against the oblivious adversary on a single machine, therefore matching the lower bound for this case.
We consider testing directed graphs for being Eulerian in the orientation model introduced in [15... more We consider testing directed graphs for being Eulerian in the orientation model introduced in [15]. Despite the local nature of the property of being Eulerian, it turns out to be significantly harder for testing than other properties studied in the orientation model. We show a non-constant lower bound on the query complexity of 2-sided tests and a linear lower bound on the query complexity of 1-sided tests for this property. On the positive side, we give several 1-sided and 2-sided tests, including a sub-linear query complexity 2-sided test for general graphs. For special classes of graphs, including bounded-degree graphs and expander graphs, we provide improved results. In particular, we give a 2-sided test with constant query complexity for dense graphs, as well as for expander graphs with a constant expansion parameter.
In this paper we establish an exponential lower bound on the size of syntactic non-deterministic ... more In this paper we establish an exponential lower bound on the size of syntactic non-deterministic read d-times branching programs for d ≤ log n/10 5 computing a class of monotone CNFs with a linear number of clauses. This result provides the first separation of classes NP and co-NP for syntactic branching programs with a logarithmic repetition and the first separation of syntactic non-deterministic branching programs with a logarithmic repetition from small monotone CNFs.
Symposium on Theoretical Aspects of Computer Science, 2017
Distribution testing deals with what information can be deduced about an unknown distribution ove... more Distribution testing deals with what information can be deduced about an unknown distribution over {1,. .. , n}, where the algorithm is only allowed to obtain a relatively small number of independent samples from the distribution. In the extended conditional sampling model, the algorithm is also allowed to obtain samples from the restriction of the original distribution on subsets of {1,. .. , n}. In 2015, Canonne, Diakonikolas, Gouleakis and Rubinfeld unified several previous results, and showed that for any property of distributions satisfying a "decomposability" criterion, there exists an algorithm (in the basic model) that can distinguish with high probability distributions satisfying the property from distributions that are far from it in variation distance. We present here a more efficient yet simpler algorithm for the basic model, as well as very efficient algorithms for the conditional model, which until now was not investigated under the umbrella of decomposable properties. Additionally, we provide an algorithm for the conditional model that handles a much larger class of properties. Our core mechanism is a way of efficiently producing an interval-partition of {1,. .. , n} that satisfies a "fine-grain" quality. We show that with such a partition at hand we can directly move forward with testing individual intervals, instead of first searching for the "correct" partition of {1,. .. , n}.
Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices ar... more Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst case guarantees. We propose a new data structure, the dynamic tree (D-tree), together with algorithms to construct and maintain it. The D-tree is the first data structure that scales to fully dynamic graphs with millions of vertices and edges and, on average, answers connectivity queries much faster than data structures with worst case guarantees.
A string α ∈ Σ n is called p-periodic, if for every i, j ∈ {1,. .. , n}, such that i ≡ j mod p, α... more A string α ∈ Σ n is called p-periodic, if for every i, j ∈ {1,. .. , n}, such that i ≡ j mod p, αi = αj, where αi is the i-th place of α. A string α ∈ Σ n is said to be period(≤ g), if there exists p ∈ {1,. .. , g} such that α is p-periodic. An property tester for period(≤ g) is a randomized algorithm, that for an input α distinguishes between the case that α is in period(≤ g) and the case that one needs to change at least-fraction of the letters of α, so that it will become period(≤ g). The complexity of the tester is the number of letter-queries it makes to the input. We study here the complexity of testers for period(≤ g) when g varies in the range 1,. .. , n 2. We show that there exists a surprising exponential phase transition in the query complexity around g = log n. That is, for every δ > 0 and for each g, such that g ≥ (log n) 1+δ , the number of queries required and sufficient for testing period(≤ g) is polynomial in g. On the other hand, for each g ≤ logn 4 , the number of queries required and sufficient for testing period(≤ g) is only poly-logarithmic in g. We also prove an exact asymptotic bound for testing general periodicity. Namely, that 1-sided error, non adaptive-testing of periodicity (period(≤ n 2)) is Θ(√ n log n) queries.
We consider testing directed graphs for being Eulerian in the orientation model introduced in [15... more We consider testing directed graphs for being Eulerian in the orientation model introduced in [15]. Despite the local nature of the property of being Eulerian, it turns out to be significantly harder for testing than other properties studied in the orientation model. We show a non-constant lower bound on the query complexity of 2-sided tests and a linear lower bound on the query complexity of 1-sided tests for this property. On the positive side, we give several 1-sided and 2-sided tests, including a sub-linear query complexity 2-sided test for general graphs. For special classes of graphs, including bounded-degree graphs and expander graphs, we provide improved results. In particular, we give a 2-sided test with constant query complexity for dense graphs, as well as for expander graphs with a constant expansion parameter.
We prove almost tight bounds on the length of paths in 2-edge-connected cubic graphs. Concretely,... more We prove almost tight bounds on the length of paths in 2-edge-connected cubic graphs. Concretely, we show that (i) every 2-edge-connected cubic graph of size n has a path of length Ω log 2 n log log n , and (ii) there exists a 2-edge-connected cubic graph, such that every path in the graph has length O(log 2 n)
Journal of Computer and System Sciences, Feb 1, 2013
The covering and boundedness problems for branching vector addition systems are shown complete fo... more The covering and boundedness problems for branching vector addition systems are shown complete for doubly-exponential time.
We study the query complexity of testing for properties defined by read once formulas, as instanc... more We study the query complexity of testing for properties defined by read once formulas, as instances of massively parametrized properties, and prove several testability and non-testability results. First we prove the testability of any property accepted by a Boolean read-once formula involving any bounded arity gates, with a number of queries exponential in ǫ, doubly exponential in the arity, and independent of all other parameters. When the gates are limited to being monotone, we prove that there is an estimation algorithm, that outputs an approximation of the distance of the input from satisfying the property. For formulas only involving And/Or gates, we provide a more efficient test whose query complexity is only quasipolynomial in ǫ. On the other hand, we show that such testability results do not hold in general for formulas over non-Boolean alphabets; specifically we construct a property defined by a read-once arity 2 (non-Boolean) formula over an alphabet of size 4, such that any 1/4-test for it requires a number of queries depending on the formula size. We also present such a formula over an alphabet of size 5 that additionally satisfies a strong monotonicity condition.
Distribution testing deals with what information can be deduced about an unknown distribution ove... more Distribution testing deals with what information can be deduced about an unknown distribution over {1,. .. , n}, where the algorithm is only allowed to obtain a relatively small number of independent samples from the distribution. In the extended conditional sampling model, the algorithm is also allowed to obtain samples from the restriction of the original distribution on subsets of {1,. .. , n}. In 2015, Canonne, Diakonikolas, Gouleakis and Rubinfeld unified several previous results, and showed that for any property of distributions satisfying a "decomposability" criterion, there exists an algorithm (in the basic model) that can distinguish with high probability distributions satisfying the property from distributions that are far from it in the variation distance. We present here a more efficient yet simpler algorithm for the basic model, as well as very efficient algorithms for the conditional model, which until now was not investigated under the umbrella of decomposable properties. Additionally, we provide an algorithm for the conditional model that handles a much larger class of properties. Our core mechanism is a way of efficiently producing an interval-partition of {1,. .. , n} that satisfies a "fine-grain" quality. We show that with such a partition at hand we can directly move forward with testing individual intervals, instead of first searching for the "correct" partition of {1,. .. , n}.
Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices ar... more Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst case guarantees. We propose a new data structure, the dynamic tree (D-tree), together with algorithms to construct and maintain it. The D-tree is the first data structure that scales to fully dynamic graphs with millions of vertices and edges and, on average, answers connectivity queries much faster than data structures with worst case guarantees.
Society for Industrial and Applied Mathematics eBooks, 2020
Please refer to published version for the most recent bibliographic citation information. If a pu... more Please refer to published version for the most recent bibliographic citation information. If a published version is known of, the repository item page linked to above, will contain details on accessing it.
1 1-sided which doesn’t work Let w ∈ {0, 1}n, i ∈ [n] and Q ⊆ [n]. We use wi to denote the i’th l... more 1 1-sided which doesn’t work Let w ∈ {0, 1}n, i ∈ [n] and Q ⊆ [n]. We use wi to denote the i’th letter of w and wQ to be the a string v ∈ {0, 1}|Q| such that, for every j ∈ [|Q|], vj = wk(j), where k(j) is the j’th smallest member of Q. Definition 1.1 (constraints, etc). A q-constraint is a pair C = (Q,U) where Q is a subset of [n] of size at most q, and U is a subset of {0, 1}|Q|. A word w ∈ {0, 1}n is said to violate C if wQ ∈ U . A q-formula F is a set of q-constraints, all of whose corresponding Q sets are distinct. The property PF is defined as the set of words that violate no member of F . We say that F is solvable if PF 6= ∅. Given a property P and a a set Q ⊆ [n], the natural constraint is CQ = (Q,UQ), where UQ ⊆ {0, 1}q is the set of strings v for which there exist no w ∈ P with wQ = v. A witness against a word w 6∈ P is a set Q so that w does not satisfy the natural constraint (Q,UQ). Similarly, given a set of subsets of [n], the corresponding natural formula is the set of...
A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T cont... more A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T controls another vertex y if there exists a directed path of length at most two from x to y. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu [8] in investigating the query complexity of finding a king, that is, the number of arcs in T one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least Ω(n 4/3) arcs and provided a strategy that queries at most O(n 3/2). While this upper bound has not yet been improved for the original problem, Biswas et al. [3] proved that with O(n 4/3) queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using O(n 4/3 polylog n) queries, we can identify a (1 2 + 2 17)-king. To achieve this goal we use a novel structural result for tournaments.
A string α ∈ Σ is called p-periodic, if for every i, j ∈ {1, . . . , n}, such that i ≡ j mod p, α... more A string α ∈ Σ is called p-periodic, if for every i, j ∈ {1, . . . , n}, such that i ≡ j mod p, αi = αj , where αi is the i-th place of α. A string α ∈ Σ is said to be period(≤ g), if there exists p ∈ {1, . . . , g} such that α is p-periodic. An property tester for period(≤ g) is a randomized algorithm, that for an input α distinguishes between the case that α is in period(≤ g) and the case that one needs to change at least -fraction of the letters of α, so that it will become period(≤ g). The complexity of the tester is the number of letter-queries it makes to the input. We study here the complexity of testers for period(≤ g) when g varies in the range 1, . . . , n 2 . We show that there exists a surprising exponential phase transition in the query complexity around g = log n. That is, for every δ > 0 and for each g, such that g ≥ (log n), the number of queries required and sufficient for testing period(≤ g) is polynomial in g. On the other hand, for each g ≤ logn 4 , the number...
We study the following question, communicated to us by Miklós Ajtai: Can all explicit (e.g., poly... more We study the following question, communicated to us by Miklós Ajtai: Can all explicit (e.g., polynomial time computable) functions f : ({0, 1} w ) 3 → {0, 1} w be computed by word circuits of constant size? Here, a word circuit is an acyclic circuit where each wire holds a word (i.e., an element of {0, 1} w ) and each gate G computes some binary operation gG : ({0, 1} w ) 2 → {0, 1} w , defined for all word lengths w. As our main result, we present an explicit function so that its w'th slice for any w ≥ 8 cannot be computed by word circuits with at most 4 gates. Also, we formally relate Ajtai's question to open problems concerning ACC 0 circuits.
Uploads
Papers by Oded Lachish