https://sublinear.info/api.php?action=feedcontributions&feedformat=atom&user=Andoni 2026-04-22T16:59:21Z User contributions MediaWiki 1.31.10 https://sublinear.info/index.php?title=Open_Problems:25&diff=1314 2019-11-08T17:02:31Z

<p>Andoni: </p> <hr /> <div>{{Header<br /> |source=kanpur09<br /> |who=T. S. Jayram<br /> }}<br /> == Poincaré Inequalities ==<br /> Alice and Bob are given two points $x$ and $y$, respectively, from a specific metric space $\mathcal M$. We are interested in deciding whether $d_{\mathcal M}(x,y) \le R$ or $d_{\mathcal M}(x,y) \ge \alpha R$, where $d_{\mathcal M}$ is the distance function of $\mathcal M$, $R &gt; 0$, and $\alpha &gt; 1$. What amount of information must be exchanged in order to solve this problem? Answering this question is interesting in any standard communication model: unrestricted communication between the players, one-way communication, sketching, etc.<br /> <br /> The above question can partially be answered if the metric satisfies a specific &amp;ldquo;gap&amp;rdquo; Poincaré inequality {{cite|AndoniJP-10}}. It is known that another kind of Poincaré inequality is equivalent to non-embeddability into $\ell_2^2$ {{cite|Matousek-02}}, but it is not known if non-embeddability into $\ell_2^2$ implies lower bounds for communication complexity. Can one show a formal connection between the communication complexity for approximating the distance between two points and non-embeddability into $\ell_2^2$?<br /> <br /> == Product Metrics ==<br /> We are also interested in the following general class of metrics. Let each $\mathcal M_i= \langle S_i, d_i\rangle$, $1 \le i \le k$, be a metric space on a set $S_i$ with a distance function $d_i$. A ''product metric space'' $\bigoplus_{i=1}^{k} \mathcal M_i$ is defined on the product $S_1 \times \ldots \times S_k$ with the distance function<br /> $$d\left((x_1,\ldots,x_k),(y_1,\ldots,y_k)\right) = \operatorname{\bf op}\left(d_1(x_1,y_1),\ldots,d_k(x_k,y_k)\right),$$<br /> where ${\bf op}$ is a symmetric operator. For instance, $\bigoplus_{i=1}^{k} \mathcal M_i$ is a proper metric space if ${\bf op}$ is the maximum operator or the $p$-th norm for any $p \in [1,\infty)$. The case when $\bigoplus_{i=1}^{k} \mathcal M_i$ is not necessarily a metric space also finds applications.<br /> <br /> Applications of product metric spaces include a nearest neighbor data structure for Ulam distance {{cite|AndoniIK-09}}, and a near-linear time subpolynomial-approximation algorithm for edit distance {{cite|AndoniO-09}}.<br /> <br /> The following questions arise in the context of product spaces:<br /> # Can one design efficient communication protocols for computing the distance between a pair of points? Suppose that there is an efficient communication protocol for each $\mathcal M_i$. What is the communication complexity for computing the distance between two points in $\bigoplus_{i=1}^{k}\mathcal M_i$? Andoni, Jayram, and Pătraşcu {{cite|AndoniJP-10}} prove lower bounds for some product metrics. Jayram and Woodruff {{cite|JayramW-09}} show streaming algorithms which yield communication protocols.<br /> # Can one design efficient streaming algorithms and data structures for product metric spaces? In particular, can one efficiently compute the distance between a pair of points? Jayram and Woodruff {{cite|JayramW-09}} consider the related question of computing ''cascaded norms''.<br /> <br /> ==Update==<br /> <br /> It has been shown {{cite|AndoniKR-14}} that for '''normed spaces''' the above implication is true: if a snowflake of a normed space does not embed into $\ell_2$ (in fact, more generally, does not ''uniformly embed'' into a Hilbert space), then, there is a non-trivial communication lower bound for distinguishing small and large distances. Furthermore, {{cite|KhotN-19}} show that a similar statement is not true for metrics in general: there exist metrics which are sketchable with constant approximation and space, but do not $O(1)$-embed into $\ell_{1-\epsilon}$ for any $\epsilon&lt;1/2$.</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:25&diff=1313 2019-11-08T17:02:13Z

<p>Andoni: /* Update */ adding KN19</p> <hr /> <div>{{Header<br /> |source=kanpur09<br /> |who=T. S. Jayram<br /> }}<br /> == Poincaré Inequalities ==<br /> Alice and Bob are given two points $x$ and $y$, respectively, from a specific metric space $\mathcal M$. We are interested in deciding whether $d_{\mathcal M}(x,y) \le R$ or $d_{\mathcal M}(x,y) \ge \alpha R$, where $d_{\mathcal M}$ is the distance function of $\mathcal M$, $R &gt; 0$, and $\alpha &gt; 1$. What amount of information must be exchanged in order to solve this problem? Answering this question is interesting in any standard communication model: unrestricted communication between the players, one-way communication, sketching, etc.<br /> <br /> The above question can partially be answered if the metric satisfies a specific &amp;ldquo;gap&amp;rdquo; Poincaré inequality {{cite|AndoniJP-10}}. It is known that another kind of Poincaré inequality is equivalent to non-embeddability into $\ell_2^2$ {{cite|Matousek-02}}, but it is not known if non-embeddability into $\ell_2^2$ implies lower bounds for communication complexity. Can one show a formal connection between the communication complexity for approximating the distance between two points and non-embeddability into $\ell_2^2$?<br /> <br /> == Product Metrics ==<br /> We are also interested in the following general class of metrics. Let each $\mathcal M_i= \langle S_i, d_i\rangle$, $1 \le i \le k$, be a metric space on a set $S_i$ with a distance function $d_i$. A ''product metric space'' $\bigoplus_{i=1}^{k} \mathcal M_i$ is defined on the product $S_1 \times \ldots \times S_k$ with the distance function<br /> $$d\left((x_1,\ldots,x_k),(y_1,\ldots,y_k)\right) = \operatorname{\bf op}\left(d_1(x_1,y_1),\ldots,d_k(x_k,y_k)\right),$$<br /> where ${\bf op}$ is a symmetric operator. For instance, $\bigoplus_{i=1}^{k} \mathcal M_i$ is a proper metric space if ${\bf op}$ is the maximum operator or the $p$-th norm for any $p \in [1,\infty)$. The case when $\bigoplus_{i=1}^{k} \mathcal M_i$ is not necessarily a metric space also finds applications.<br /> <br /> Applications of product metric spaces include a nearest neighbor data structure for Ulam distance {{cite|AndoniIK-09}}, and a near-linear time subpolynomial-approximation algorithm for edit distance {{cite|AndoniO-09}}.<br /> <br /> The following questions arise in the context of product spaces:<br /> # Can one design efficient communication protocols for computing the distance between a pair of points? Suppose that there is an efficient communication protocol for each $\mathcal M_i$. What is the communication complexity for computing the distance between two points in $\bigoplus_{i=1}^{k}\mathcal M_i$? Andoni, Jayram, and Pătraşcu {{cite|AndoniJP-10}} prove lower bounds for some product metrics. Jayram and Woodruff {{cite|JayramW-09}} show streaming algorithms which yield communication protocols.<br /> # Can one design efficient streaming algorithms and data structures for product metric spaces? In particular, can one efficiently compute the distance between a pair of points? Jayram and Woodruff {{cite|JayramW-09}} consider the related question of computing ''cascaded norms''.<br /> <br /> ==Update==<br /> <br /> It has been shown {{cite|AndoniKR-14}} that for '''normed spaces''' the above implication is true: if a snowflake of a normed space does not embed into $\ell_2$ (in fact, more generally, does not ''uniformly embed'' into a Hilbert space), then, there is a non-trivial communication lower bound for distinguishing small and large distances. Furtermore, {{cite|KhotN-19}} show that a similar statement is not true for metrics in general: there exist metrics which are sketchable with constant approximation and space, but do not $O(1)$-embed into $\ell_{1-\epsilon}$ for any $\epsilon&lt;1/2$.</div>

Andoni https://sublinear.info/index.php?title=Bibliography&diff=1312 2019-11-08T16:59:28Z

<p>Andoni: </p> <hr /> <div>&lt;p class=&quot;dontprint&quot;&gt;'''Citations:''' Write &lt;tt&gt;&amp;#123;&amp;#123;cite&amp;#124;&lt;/tt&gt;''paper_id_1''&lt;tt&gt;&amp;#124;&lt;/tt&gt;''paper_id_2''&lt;tt&gt;&amp;#124;&lt;/tt&gt;&amp;hellip;&lt;tt&gt;&amp;#124;&lt;/tt&gt;''paper_id_k''&lt;tt&gt;&amp;#125;&amp;#125;&lt;/tt&gt; to cite papers ''paper_id_1'', ''paper_id_2'', &amp;hellip;, ''paper_id_k''. For instance, &lt;tt&gt;&amp;#123;&amp;#123;cite&amp;#124;AlonMS-99&amp;#124;BlumLR-93&amp;#125;&amp;#125;&lt;/tt&gt; results in {{cite|AlonMS-99|BlumLR-93}}.&lt;/p&gt;<br /> &lt;hr class=&quot;dontprint&quot;&gt;<br /> <br /> {{bibentry|AaronsonW-09|Scott Aaronson and Avi Wigderson. ''Algebrization: A New Barrier in Complexity Theory.'' ACM Transactions on Computation Theory, 1(1), 2009.}}<br /> <br /> {{bibentry|AcharyaCK-14|Jayadev Acharya, Clément Canonne, and Gautam Kamath. ''A Chasm Between Identity and Equivalence Testing with Conditional Queries.'' 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Andoni https://sublinear.info/index.php?title=Open_Problems:48&diff=580 2012-12-13T16:50:02Z

<p>Andoni: </p> <hr /> <div>{{Header<br /> |title=Sketching Shift Metrics<br /> |source=bertinoro11<br /> |who=Alexandr Andoni<br /> }}<br /> For any $x,y \in \{0,1\}^n$, define the ''shift metric''<br /> \[\operatorname{sh}(x,y)=\min_{ \sigma} H(x, \sigma(y)), \]<br /> where $\sigma$ ranges over all $n$ cyclic permutations of $\{1 \ldots<br /> n\}$, and $H()$ is the hamming distance.<br /> <br /> For any $c&gt;20$, the promise problem $P_c$ is to distinguish whether $\operatorname{sh}(x,y)&gt;n/10$ or $\operatorname{sh}(x,y)&lt;n/c$.<br /> Consider probabilistic mappings $L_c: \{0,1\}^n \to \{0,1\}^s$.<br /> We say that $L_c$ is a sketching scheme for $P_c$ if there is an algorithm that, for any $x,y \in \{0,1\}^n$ satisfying the promise of $P_c$, given $L_c(x)$ and $L_c(y)$, solves $P_c$ with probability at least $0.9$.<br /> <br /> '''Question:''' Is there a sketching scheme for $P_c$ where $c=O(1)$ and $s=O(1)$?<br /> <br /> '''Background:''' If the shift metric is replaced by Hamming metric, one can achieve $s=O(1)$ using random sampling {{cite|KushilevitzOR-00}}. The actual problem can be solved for $c=O(\log^2 n)$ and $s=O(1)$ {{cite|AndoniIK-08}}. The algorithm proceeds by embedding the shift metric into Hamming metrics, and it is known that this step must induce $\Omega(\log n)$ distortion {{cite|KhotN-06}}. There's also a solution for $c=1+\epsilon$ and $s=\tilde{O}(\epsilon^{-2}\sqrt{n})$ {{cite|CrouchM-11}}.</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:51&diff=545 2012-12-12T02:28:09Z

<p>Andoni: </p> <hr /> <div>{{Header<br /> |title=&quot;For all&quot; guarantee for computationally bounded adversaries<br /> |source=dortmund12<br /> |who=Martin Strauss<br /> }}<br /> There are two types of compressed sensing guarantees, illustrated<br /> using two players:<br /> <br /> * ''For all'': Charlie constructs the sensing matrix $\phi$, and then Mallory constructs the signal $x=x(\phi)$ as a function of $\phi$. The Compressed Sensing question is to recover the approximate signal $\tilde x$ from the measurement $\phi x$. The best guarantee possible is the following $\ell_2/\ell_1$ guarantee:$$||\tilde x - x||_2 \le \epsilon/\sqrt{k} ||x_{opt} - x||_1.$$<br /> <br /> * ''For each'': Charlie construct a distribution $D$ over sensing matrices $\phi$. Then Mallory constructs a vector $x=x(D)$ dependent on the distribution only. Finally, a sensing matrix $\phi$ is sampled from the distribution $D$. The goal is again to recover $\tilde x$, with good probability over the choice of $\phi$. It turns out a stronger guarantee, termed $\ell_2/\ell_2$, is possible: $$||\tilde x - x||_2 \le (1+\epsilon)||x_{opt} - x||_2.$$<br /> <br /> In some sense the two &quot;worlds&quot; are incomparable: the first one works<br /> for all $x$ but obtains weaker error guarantee, and the second one<br /> works for each $x$ with some probability but gets better error guarantee. <br /> <br /> '''Question is''': How can we get the best of both worlds (&quot;for all&quot; with<br /> $\ell_2/\ell_2$ error) ?<br /> <br /> Once we require &quot;for all&quot;, it is provably impossible to obtain $\ell_2/\ell_2$ guarantee. But what if Mallory has bounded computational resources to construct a &quot;bad&quot; $x$?<br /> <br /> A preliminary result considers the following setting. Mallory sees $\phi$ and writes down a sketch of $\phi$ (in bounded space). Then Mallory produces $x$ from this sketch only. Then $\ell_2/\ell_2$ is<br /> possible for such $x$'s.<br /> <br /> Generally, we would like to allow Mallory to be probabilistic polynomial time, and have a $\phi$ so that Mallory still cannot find an input $x=x(\phi)$ that breaks the recovery algorithm.</div>

Andoni https://sublinear.info/index.php?title=Bibliography&diff=544 2012-12-12T02:26:40Z

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Andoni https://sublinear.info/index.php?title=Open_Problems:60&diff=541 2012-12-12T02:19:25Z

<p>Andoni: Created page with &quot;{{Header |title=Single-Pass Unweighted Matchings |source=dortmund12 |who=Andrew McGregor }} Suppose you have $O(n \hbox{polylog} n)$ memory and a single pass over a stream of ...&quot;</p> <hr /> <div>{{Header<br /> |title=Single-Pass Unweighted Matchings<br /> |source=dortmund12<br /> |who=Andrew McGregor<br /> }}<br /> Suppose you have $O(n \hbox{polylog} n)$ memory and a single pass over a stream of $m$ edges (arbitrarily ordered) on $n$ nodes. How well can you approximate the size of the maximum cardinality matching? A trivial greedy algorithm finds a $1/2$-approximation but that's still the best known algorithm in the general setting. Kapralov {{cite|Kapralov-12}} showed that achieving better than a $1-1/e$ approximation is impossible. If the stream is randomly ordered, Konrad et al. {{cite|KonradMM-12}} presented a $1/2 + 0.005$-approximation. Other variants of the question are also open, e.g., achieving a $(1-\epsilon)$ approximation with multiple passes (see, e.g., Ahn and Guha {{cite|AhnG-11}}) or the best approximation possible for maximum weighted matching in a single pass (see, e.g., Epstein et al. {{cite|EpsteinLMS-11}}).</div>

Andoni https://sublinear.info/index.php?title=Bibliography&diff=538 2012-12-12T02:15:58Z

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Andoni https://sublinear.info/index.php?title=Open_Problems:59&diff=536 2012-12-12T02:12:25Z

<p>Andoni: Created page with &quot;{{Header |title=Low Expansion Encoding of Edit Distance |source=dortmund12 |who=Hossein Jowhari }} Let $T = \bigcup_{i=1}^{n} \{0,1\}^i$. For pair of strings $(x,y) \in T \tim...&quot;</p> <hr /> <div>{{Header<br /> |title=Low Expansion Encoding of Edit Distance<br /> |source=dortmund12<br /> |who=Hossein Jowhari<br /> }}<br /> Let $T = \bigcup_{i=1}^{n} \{0,1\}^i$. For pair of strings $(x,y) \in T \times T$ let $ed(x,y)$ denote the edit distance between $x$ and $y$ which is defined as the minimum number of character insertion, deletion and substitution needed for converting $x$ into $y$. <br /> <br /> '''Question''': is there a mapping $f:T \rightarrow \{0,1\}^{m}$ satisfying the following conditions<br /> * $f$ is injective, i.e. it does not map different inputs to the same point.<br /> * $m=O(n^c)$ for some constant $c \geq 1$.<br /> * For strings with $ed(x,y)=1$ we have $\mathcal{H}(f(x),f(y)) \le C$ for $C=o(\log n)$.<br /> <br /> The same question holds for randomized mappings as long as they map different $x$ and $y$ to different points with high probability. Currently the best upper bound on $C$ is $O(\log n\log^*n)$ achieved through a randomized mapping that deploys the Locally Consistent Parsing method {{cite|CormodePSV-00}}. For non-repetitive strings (the Ulam distance) there is a deterministic mapping with $C\leq 6$ and $c=2$. Preferably we would like to have mappings that are efficiently computable and are equipped with polynomial time decoding algorithms ($x$ can be obtained from $f(x)$ efficiently). See {{cite|Jowhari-12}} for motivations on the problem.</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:58&diff=535 2012-12-12T02:10:06Z

<p>Andoni: Created page with &quot;{{Header |title=Signatures for set equality |source=dortmund12 |who=Rasmus Pagh }} Given $S\subseteq\{1,..n\}$, we would like to construct a fingerprint so that later, given f...&quot;</p> <hr /> <div>{{Header<br /> |title=Signatures for set equality<br /> |source=dortmund12<br /> |who=Rasmus Pagh<br /> }}<br /> Given $S\subseteq\{1,..n\}$, we would like to construct a fingerprint so that later, given fingerprints of two sets, we can check the equality of the two sets.<br /> <br /> There are (at least) two possible solutions to the problem:<br /> * $h(S)=\sum_{i\in S} x^i\mod p$ for random $x\in \Z_p$. Update time would be roughly $\log p=\Omega(\log n)$. One would like to obtain a better update time.<br /> * $h(S)=\prod_{i\in S} (x-i) \mod p$, and random $x$. Insertion can be done in constant time. But the fingerprint is not linear.<br /> <br /> '''Question''': Can we construct a fingerprint that achieves constant update time and is linear, while using $O(\log n)$ random bits? Ideally updates would include insertions and deletions.</div>

Andoni https://sublinear.info/index.php?title=Bibliography&diff=534 2012-12-12T02:06:45Z

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Andoni https://sublinear.info/index.php?title=Open_Problems:57&diff=532 2012-12-12T02:02:11Z

<p>Andoni: Created page with &quot;{{Header |title=Coding theory in the streaming model |source=dortmund12 |who=Atri Rudra }} Consider the problem of &quot;codeword testing&quot; in the data stream model. In particular, ...&quot;</p> <hr /> <div>{{Header<br /> |title=Coding theory in the streaming model<br /> |source=dortmund12<br /> |who=Atri Rudra<br /> }}<br /> Consider the problem of &quot;codeword testing&quot; in the data stream model. In particular, consider a code<br /> \[C:\Sigma^k\rightarrow\Sigma^n,\]<br /> with distance $d$. Recall that the distance of a code $C$ is the minimum Hamming distance between any two codewords, i.e. $\min_{\mathbb{x}\neq \mathbb{y}\in\Sigma^k} |\{i\in [n]| C(\mathbb{x})_i\neq C(\mathbb{y})_i\}|$. The specific problem is the following<br /> The input to the problem is a vector $\mathbb{y}\in\Sigma^n$ and integer parameters $0\le \tau_1&lt;\tau_2\le n$. The algorithm has to decide whether<br /> \[\Delta(\mathbb{y},C)\le \tau_1~ \mathrm{ or }~ \Delta(\mathbb{y},C)\geq \tau_2.\]<br /> $\Delta(\mathbb{y},C)$ is the Hamming distance of $\mathbb{y}$ from the closest codeword in $C$.<br /> <br /> Ideally, we want a one-pass, $\log^{O(1)}{n}$ space algorithm to solve the problem above for some ''good'' code $C$ (that is, we have $k\ge \Omega(n)$ and $d\ge \Omega(n)$). Or if we prove a hardness result, one would like a hardness result for ''every'' good code $C$. (For the sake of simplicity, assume that the algorithm has access to some succinct description of the code $C$.)<br /> <br /> The main technical motivation comes from the case when $\tau_1=0$ and $\tau_2\ge \epsilon n$ for any fixed $\epsilon&gt;0$ but with ''constant'' number of queries to $\mathbb{y}$ (i.e. in the property testing world). This question is perhaps the open question in the codeword testing literature. The case of $\tau_1&gt;0$ also makes sense in the property testing world and has been studied {{cite|GuruswamiR-05}}. (See the paper for some potential practical motivations.)<br /> <br /> One of the original motivation (in {{cite|RudraU-10}}) for the study of the data-streaming version of the question was possibly to use communication complexity results to prove the impossibility of good locally testable codes.<br /> <br /> It was shown in {{cite|RudraU-10}} that for the well-known Reed-Solomon codes, the data stream version of the problem can be solved for $\tau_1=0$ and $\tau_2=1$ with one pass and logarithmic space. It can also be shown that the classical Berlekamp-Massey algorithm for decoding Reed-Solomon codes implies a solution for the case $\tau_2=\tau_1+1$ with one pass and space $\tilde{O}(\tau_1)$. (There is a small catch: the algorithm actually computes the location of errors ''if'' the number of errors is at most $\tau_1$. However, results in {{cite|RudraU-10}} can be used to verify if the returned error locations are indeed correct.)<br /> <br /> Finally, {{cite|McGregorRU-11}} showed how to solve this problem in one pass and $O(k\log{n})$ space.<br /> This question is wide open:<br /> Solve the problem above with one pass and $\tilde{O}(\min(k,\tau_1))$ space.<br /> <br /> In fact the very special case of the problem above for $k=\tau_1=\sqrt{n}$ with one pass and space $o(\sqrt{n})$ is also open. This is open even for the special case of Reed-Solomon codes.</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:56&diff=530 2012-12-12T01:51:03Z

<p>Andoni: Created page with &quot;{{Header |title=Efficient measures of ''surprisingness'' of sequences |source=dortmund12 |who=Rina Panigrahy }} Consider a sequence of iid random bits $S\in\{0,1\}^n$. '''Que...&quot;</p> <hr /> <div>{{Header<br /> |title=Efficient measures of ''surprisingness'' of sequences<br /> |source=dortmund12<br /> |who=Rina Panigrahy<br /> }}<br /> Consider a sequence of iid random bits $S\in\{0,1\}^n$.<br /> <br /> '''Question''': Find efficient measures of how surprising/unbelievable $S$<br /> appears to be. (Good heuristic for measuring how probable/improbably a<br /> string is.)<br /> <br /> For example, if we see $0,0,0,\ldots $, we won't believe it is random<br /> (i.e., it is surprising.) <br /> <br /> One existing measure is the ($k^{th}$-order) Shannon entropy $H_k$ ($H_0$<br /> would correspond to taking the entropy of the empirical frequencies of<br /> 0s and 1s). However, it fails to say that a string like<br /> $(0,0,...0,1,1,...1)$ is surprising (from the point of view of<br /> densities it looks pretty random).<br /> <br /> Ideal solution is to consider the Kolmogorov complexity, but it is<br /> hard (impossible) to compute.<br /> <br /> A particular setting of the strings to consider may be: suppose each<br /> bit is generated from a biased independent coin, but the bias of the<br /> coin changes (slowly?) over time. Is there a good compression here?</div>

Andoni https://sublinear.info/index.php?title=Bibliography&diff=528 2012-12-12T01:49:38Z

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Andoni https://sublinear.info/index.php?title=Open_Problems:55&diff=525 2012-12-12T01:48:40Z

<p>Andoni: </p> <hr /> <div>{{Header<br /> |title=Applications of Clifford Algebras in Graph Streams<br /> |source=dortmund12<br /> |who=He Sun<br /> }}<br /> Some of the recent results in graph streaming algorithms {{cite|KaneMSS-12|ManjunathMPS-11}} use ''complex-valued'' sketches to capture the graph structure. While it had been known earlier that integer-valued sketches can be used to count triangles, Kane et al. {{cite|KaneMSS-12}} developed a complex-valued sketch to count the number of occurrences of an arbitrary subgraph of constant size. These techniques also extend to variations of the subgraph counting problem, for instance counting a directed or (labelled) subgraph. However, the bounds on the space complexity which depends on the variance of the sketches are quite loose for most graph families.<br /> <br /> It is interesting to compare these results to the framework of designing randomized algorithms for computing the permanent. Let $A$ be a 0-1 matrix, and $B$ be the matrix obtained from $A$ by replacing each 1 uniformly and randomly with an element from a finite set $D$. With suitable choices of the set $D$, the determinant of $B$ can be used to approximate the permanent of $A$. As shown by Chien et al. {{cite|ChienRS-03}} and discussed by Muthukrishnan {{cite|Muthukrishnan-06}}, by choosing elements of $D$ from $\mathbf{Z}$, $\mathbf{C}$, or a Clifford algebra $\mathbf{CL}$, the variance of the estimator drops significantly each time when we move to a more &quot;complex&quot; algebra. It seems plausible that similar techniques can be used to improve the space complexity of graph streaming algorithms which are based on complex-valued random variables.<br /> <br /> '''Question''': Find suitable applications of Clifford algebra in designing algorithms in graph streams.</div>

Andoni https://sublinear.info/index.php?title=Bibliography&diff=524 2012-12-12T01:42:55Z

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Andoni https://sublinear.info/index.php?title=Open_Problems:55&diff=523 2012-12-12T01:40:36Z

<p>Andoni: Created page with &quot;{{Header |title=Applications of Clifford Algebras in Graph Streams |source=dortmund12 |who=He Sun }} Some of the recent results in graph streaming algorithms {{cite|KaneMSS-12...&quot;</p> <hr /> <div>{{Header<br /> |title=Applications of Clifford Algebras in Graph Streams<br /> |source=dortmund12<br /> |who=He Sun<br /> }}<br /> Some of the recent results in graph streaming algorithms {{cite|KaneMSS-12|ManjunathMPS-11}} use ''complex-valued'' sketches to capture the graph structure. While it had been known earlier that integer-valued sketches can be used to count triangles, Kane et al. {{cite|KaneMSS-12}} developed a complex-valued sketch to count the number of occurrences of an arbitrary subgraph of constant size. These techniques also extend to variations of the subgraph counting problem, for instance counting a directed or (labelled) subgraph. However, the bounds on the space complexity which depends on the variance of the sketches are quite loose for most graph families.<br /> <br /> It is interesting to compare these results to the framework of designing randomized algorithms for computing the permanent. Let $A$ be a 0-1 matrix, and $B$ be the matrix obtained from $A$ by replacing each 1 uniformly and randomly with an element from a finite set $D$. With suitable choices of the set $D$, the determinant of $B$ can be used to approximate the permanent of $A$. As shown by Chien et al. {{cite|ChienRS-03}} and discussed by Muthukrishnan {{cite|Muthukrishnan-05}}, by choosing elements of $D$ from $\mathbf{Z}$, $\mathbf{C}$, or a Clifford algebra $\mathbf{CL}$, the variance of the estimator drops significantly each time when we move to a more &quot;complex&quot; algebra. It seems plausible that similar techniques can be used to improve the space complexity of graph streaming algorithms which are based on complex-valued random variables.<br /> <br /> '''Question''': Find suitable applications of Clifford algebra in designing algorithms in graph streams.</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:54&diff=522 2012-12-12T01:34:38Z

<p>Andoni: Created page with &quot;{{Header |title=Faster JL Dimensionality Reduction |source=dortmund12 |who=Jelani Nelson }} The standard Johnson-Lindenstrauss lemma states the following: for any $0&lt;\epsilon&lt;...&quot;</p> <hr /> <div>{{Header<br /> |title=Faster JL Dimensionality Reduction<br /> |source=dortmund12<br /> |who=Jelani Nelson<br /> }}<br /> The standard Johnson-Lindenstrauss lemma states the following: for any $0&lt;\epsilon&lt;1/2$, any $x_1\ldots x_n\in \R^d$, there exists $A\in \R^{k\times d}$ with $k=O(1/\epsilon^2\cdot \log n)$, such that for any $i,j$ we have $\|Ax_i-Ax_j\|_2=(1\pm \epsilon)\|x_i-x_j\|_2$.<br /> <br /> The main question is to construct $A$'s that admit faster computation time of $Ax$. There are several directions to try to obtain more efficient $A$:<br /> * Fast JL (FFT-based). Here, the runtime is of the form $O(d\log d + \hbox{poly}(k))$ to compute $Ax$ ($d\log d$ is usually the most significant term).<br /> * Sparse JL. Here, the runtime is of the form $O(\epsilon k\|x\|_0+k)$, where $\|x\|_0$ is the number of non-zero coordinates of $x$ (i.e., it works well for sparse vectors).<br /> <br /> '''Question''': Can one obtain a JL matrix $A$, such that one can compute $Ax$ in time $\tilde O(\|x\|_0+k)$ ?<br /> <br /> One possible avenue would be by considering a &quot;random&quot; $k$ by $k$ submatrix of the FFT matrix. This may or may not lead to the desired result.</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:53&diff=521 2012-12-12T01:32:38Z

<p>Andoni: Created page with &quot;{{Header |title=Homomorphic hash functions |source=dortmund12 |who=Ely Porat }} '''Question''': to construct a hash function $ h:\F_p^n \to \F_p^m $, where $m&lt;n$, satisfying ...&quot;</p> <hr /> <div>{{Header<br /> |title=Homomorphic hash functions<br /> |source=dortmund12<br /> |who=Ely Porat<br /> }}<br /> '''Question''': to construct a hash function <br /> $<br /> h:\F_p^n \to \F_p^m<br /> $, where $m&lt;n$, satisfying the following properties:<br /> * h is linear: $h(u+v)=h(u)+h(v)$ for all $u,v\in \F_p^n$;<br /> * for any $u,v$, we have $\Pr_h[h(u)=h(v)]=\frac{c}{p^m}$ for some constant $c$ independent of $n,m$.<br /> <br /> One solution is by considering a random linear function, given by the matrix $M$. Then we have that $\Pr_M[Mu=Mv]=\Pr_M[M(u-v)=0]=1/p^m$. This function would require $O(nm\log p)$ random bits, and computing $h$ takes $O(nm)$ time. We would like more efficient solutions.<br /> <br /> Ely and coauthors claim a solution with $O((n+m)\log p)$ bits, and $O((n+m)\log (n+m))$ time.<br /> <br /> If one considers Reed-Salomon codes, it seems that they would give worse bound on second property (probability of collision).</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:52&diff=520 2012-12-12T01:30:51Z

<p>Andoni: Created page with &quot;{{Header |title=TSP in the streaming model |source=dortmund12 |who=Christian Sohler }} We have $n$ points living in $\{1,\ldots,\Delta\}^2$ space. '''Question''': Can we appr...&quot;</p> <hr /> <div>{{Header<br /> |title=TSP in the streaming model<br /> |source=dortmund12<br /> |who=Christian Sohler<br /> }}<br /> We have $n$ points living in $\{1,\ldots,\Delta\}^2$ space.<br /> <br /> '''Question''': Can we approximate the value of the TSP tour (Traveling Salesman<br /> Problem) of the $n$ points when streaming over the points in one pass, using<br /> small space ($\log^{O(1)}\Delta$).<br /> <br /> One can achieve a $2-$approximation by computing a minimum spanning tree<br /> in small space, and use the MST to approximate TSP. The question is<br /> whether one can obtain an approximation factor $c &lt; 2$ in polylog space?<br /> <br /> There are other natural related question, such as computing the<br /> Earth-Mover Distance over the points in the stream (has appeared previously as [[Open_Problems:49|Problem 49]]).</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:51&diff=519 2012-12-12T01:25:12Z

<p>Andoni: </p> <hr /> <div>{{Header<br /> |title=&quot;For all&quot; guarantee for computationally bounded adversaries<br /> |source=dortmund12<br /> |who=Martin Strauss<br /> }}<br /> There are two types of compressed sensing guarantees, illustrated<br /> using two players:<br /> <br /> ''For all'': Charlie constructs the sensing matrix $\phi$, and then<br /> Mallory constructs the signal $x=x(\phi)$ as a function of $\phi$. The<br /> Compressed Sensing question is to recover the approximate signal $\tilde x$ from the measurement $\phi x$. The best guarantee possible is the following $\ell_2/\ell_1$ guarantee:<br /> $$<br /> ||\tilde x - x||_2 \le \epsilon/\sqrt{k} ||x_{opt} - x||_1.<br /> $$<br /> <br /> ''For each'': Charlie construct a distribution $D$ over sensing<br /> matrices $\phi$. Then Mallory constructs a vector $x=x(D)$ dependent<br /> on the distribution only. Finally, a sensing matrix $\phi$ is<br /> sampled from the distribution $D$. The goal is again to recover<br /> $\tilde x$, with good probability over the choice of $\phi$. It<br /> turns out a stronger guarantee, termed $\ell_2/\ell_2$, is possible:<br /> $$<br /> ||\tilde x - x||_2 \le (1+\epsilon)||x_{opt} - x||_2<br /> $$<br /> <br /> In some sense the two &quot;worlds&quot; are incomparable: the first one works<br /> for all $x$ but obtains weaker error guarantee, and the second one<br /> works for each $x$ with some probability but gets better error guarantee. <br /> <br /> '''Question is''': How can we get the best of both worlds (&quot;for all&quot; with<br /> $\ell_2/\ell_2$ error) ?<br /> <br /> Once we require &quot;for all&quot;, it is provably impossible to obtain<br /> $\ell_2/\ell_2$ guarantee. But what if Mallory has bounded<br /> computational resources to construct a &quot;bad&quot; $x$?<br /> <br /> A preliminary result considers the following setting. Mallory sees<br /> $\phi$ and writes down a sketch of $\phi$ (in bounded space). Then<br /> Mallory produces $x$ from this sketch only. Then $\ell_2/\ell_2$ is<br /> possible for such $x$'s.<br /> <br /> Generally, we would like to allow Mallory to be probabilistic<br /> polynomial time, and have a $\phi$ so that Mallory still cannot find<br /> an input $x=x(\phi)$ that breaks the recovery algorithm.</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:51&diff=518 2012-12-12T01:24:55Z

<p>Andoni: </p> <hr /> <div>{{Header<br /> |title=``For all'' guarantee for computationally bounded adversaries<br /> |source=dortmund12<br /> |who=Martin Strauss<br /> }}<br /> There are two types of compressed sensing guarantees, illustrated<br /> using two players:<br /> <br /> ''For all'': Charlie constructs the sensing matrix $\phi$, and then<br /> Mallory constructs the signal $x=x(\phi)$ as a function of $\phi$. The<br /> Compressed Sensing question is to recover the approximate signal $\tilde x$ from the measurement $\phi x$. The best guarantee possible is the following $\ell_2/\ell_1$ guarantee:<br /> $$<br /> ||\tilde x - x||_2 \le \epsilon/\sqrt{k} ||x_{opt} - x||_1.<br /> $$<br /> <br /> ''For each'': Charlie construct a distribution $D$ over sensing<br /> matrices $\phi$. Then Mallory constructs a vector $x=x(D)$ dependent<br /> on the distribution only. Finally, a sensing matrix $\phi$ is<br /> sampled from the distribution $D$. The goal is again to recover<br /> $\tilde x$, with good probability over the choice of $\phi$. It<br /> turns out a stronger guarantee, termed $\ell_2/\ell_2$, is possible:<br /> $$<br /> ||\tilde x - x||_2 \le (1+\epsilon)||x_{opt} - x||_2<br /> $$<br /> <br /> In some sense the two &quot;worlds&quot; are incomparable: the first one works<br /> for all $x$ but obtains weaker error guarantee, and the second one<br /> works for each $x$ with some probability but gets better error guarantee. <br /> <br /> '''Question is''': How can we get the best of both worlds (&quot;for all&quot; with<br /> $\ell_2/\ell_2$ error) ?<br /> <br /> Once we require &quot;for all&quot;, it is provably impossible to obtain<br /> $\ell_2/\ell_2$ guarantee. But what if Mallory has bounded<br /> computational resources to construct a &quot;bad&quot; $x$?<br /> <br /> A preliminary result considers the following setting. Mallory sees<br /> $\phi$ and writes down a sketch of $\phi$ (in bounded space). Then<br /> Mallory produces $x$ from this sketch only. Then $\ell_2/\ell_2$ is<br /> possible for such $x$'s.<br /> <br /> Generally, we would like to allow Mallory to be probabilistic<br /> polynomial time, and have a $\phi$ so that Mallory still cannot find<br /> an input $x=x(\phi)$ that breaks the recovery algorithm.</div>

Andoni https://sublinear.info/index.php?title=Open_Problems:51&diff=517 2012-12-12T01:22:17Z

<p>Andoni: Created page with &quot;{{Header |title=``For all'' guarantee for computationally bounded adversaries |source=dortmund12 |who=Martin Strauss }} There are two types of compressed sensing guarantees,...&quot;</p> <hr /> <div>{{Header<br /> |title=``For all'' guarantee for computationally bounded adversaries<br /> |source=dortmund12<br /> |who=Martin Strauss<br /> }}<br /> <br /> <br /> There are two types of compressed sensing guarantees, illustrated<br /> using two players:<br /> <br /> ''For all'': Charlie constructs the sensing matrix $\phi$, and then<br /> Mallory constructs the signal $x=x(\phi)$ as a function of $\phi$. The<br /> Compressed Sensing question is to recover the approximate signal $\tilde x$ from the measurement $\phi x$. The best guarantee possible is the following $\ell_2/\ell_1$ guarantee:<br /> $$<br /> ||\tilde x - x||_2 \le \epsilon/\sqrt{k} ||x_{opt} - x||_1.<br /> $$<br /> <br /> ''For each'': Charlie construct a distribution $D$ over sensing<br /> matrices $\phi$. Then Mallory constructs a vector $x=x(D)$ dependent<br /> on the distribution only. Finally, a sensing matrix $\phi$ is<br /> sampled from the distribution $D$. The goal is again to recover<br /> $\tilde x$, with good probability over the choice of $\phi$. It<br /> turns out a stronger guarantee, termed $\ell_2/\ell_2$, is possible:<br /> $$<br /> ||\tilde x - x||_2 \le (1+\epsilon)||x_{opt} - x||_2<br /> $$<br /> <br /> In some sense the two &quot;worlds&quot; are incomparable: the first one works<br /> for all $x$ but obtains weaker error guarantee, and the second one<br /> works for each $x$ with some probability but gets better error guarantee. <br /> <br /> '''Question is''': How can we get the best of both worlds (&quot;for all&quot; with<br /> $\ell_2/\ell_2$ error) ?<br /> <br /> Once we require &quot;for all&quot;, it is provably impossible to obtain<br /> $\ell_2/\ell_2$ guarantee. But what if Mallory has bounded<br /> computational resources to construct a &quot;bad&quot; $x$?<br /> <br /> A preliminary result considers the following setting. Mallory sees<br /> $\phi$ and writes down a sketch of $\phi$ (in bounded space). Then<br /> Mallory produces $x$ from this sketch only. Then $\ell_2/\ell_2$ is<br /> possible for such $x$'s.<br /> <br /> Generally, we would like to allow Mallory to be probabilistic<br /> polynomial time, and have a $\phi$ so that Mallory still cannot find<br /> an input $x=x(\phi)$ that breaks the recovery algorithm.</div>

Andoni