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We consider the problem of single link failure in an elastic optical network, (also known as flex-grid WDM network). The task is to reroute optical connections that go through the broken link using free capacity of other links of the network. Nowadays, dynamic restoration gains popularity, in which the possiblity of rerouting is only inspected after a link failure is detected. Since the problem of recovery is NP-hard, heuristic algorithms are used to either find such routes, or suggest that the routes do not exist. In order to understand the quality of these heuristics, often mixed integer linear programming is used to obtain exact positive and negative answers. We present a detailed such model that checks whether restoration is possible without the use of additional regenerators. This means, that the new light paths need to satisfy a length constraint. As preprossing we apply a trimming procedure that takes advantage of this length constraint, and significantly speeds up the evaluation of these models. Our model is more general, and besides solving the problem of link restoration, also solves the full problem of wavelength and spectrum assignment., Comment: Accepted for IEEE BlackSeaCom conference, 7 pages

We study connections between expansion in bipartite graphs and efficient online matching modeled via several games. In the basic game, an opponent switches {\em on} and {\em off} nodes on the left side and, at any moment, at most $K$ nodes may be on. Each time a node is switched on, it must be irrevocably matched with one of its neighbors. A bipartite graph has $e$-expansion up to $K$ if every set $S$ of at most $K$ left nodes has at least $e\#S$ neighbors. If all left nodes have degree $D$ and $e$ is close to $D$, then the graph is a lossless expander. We show that lossless expanders allow for a polynomial time strategy in the above game, and, furthermore, with a slight modification, they allow a strategy running in time $O(D \log N)$, where $N$ is the number of left nodes. Using this game and a few related variants, we derive applications in data structures and switching networks. Namely, (a) 1-query bitprobe storage schemes for dynamic sets (previous schemes work only for static sets),(b) explicit space- and time-efficient storage schemes for static and dynamic sets with non-adaptive access to memory (the first fully dynamic dictionary with non-adaptive probing using almost optimal space), and (c) non-explicit constant depth non-blocking $N$-connectors with poly$(\log N)$ time path finding algorithms whose size is optimal within a factor of $O(\log N)$ (previous connectors are double-exponentially slower)., Comment: The exposition has been improved and a few minor issues have been fixed

There is a parallelism between Shannon information theory and algorithmic information theory. In particular, the same linear inequalities are true for Shannon entropies of tuples of random variables and Kolmogorov complexities of tuples of strings (Hammer et al., 1997), as well as for sizes of subgroups and projections of sets (Chan, Yeung, Romashchenko, Shen, Vereshchagin, 1998--2002). This parallelism started with the Kolmogorov-Levin formula (1968) for the complexity of pairs of strings with logarithmic precision. Longpr\'e (1986) proved a version of this formula for space-bounded complexities. In this paper we prove an improved version of Longpr\'e's result with a tighter space bound, using Sipser's trick (1980). Then, using this space bound, we show that every linear inequality that is true for complexities or entropies, is also true for space-bounded Kolmogorov complexities with a polynomial space overhead., Comment: [Better bound using improvements by Bruno Bauwens]

We consider the notion of information distance between two objects $x$ and $y$ introduced by Bennett, G\'acs, Li, Vit\'anyi, and Zurek in 1998 as the minimal length of a program that computes $x$ from $y$ as well as computing $y$ from $x$. In this paper, it was proven that the distance is equal to $\max (K(x|y),K(y|x))$ up to additive logarithmic terms, and it was conjectured that this could not be improved to $O(1)$ precision. We revisit subtle issues in the definition and prove this conjecture. We show that if the distance is at least logarithmic in the length, then this equality does hold with $O(1)$ precision for strings of equal length. Thus for such strings, both the triangle inequality and the characterization hold with optimal precision. Finally, we extend the result to sets $S$ of bounded size. We show that for each constant~$s$, the shortest program that prints an $s$-element set $S \subseteq \{0,1\}^n$ given any of its elements, has length at most $\max_{w \in S} K(S|w) + O(1)$, provided this maximum is at least logarithmic in~$n$., Comment: arXiv admin note: substantial text overlap with arXiv:1807.11087

We put forth new models for universal channel coding. Unlike standard codes which are designed for a specific type of channel, our most general universal code makes communication resilient on every channel, provided the noise level is below the tolerated bound, where the noise level t of a channel is the logarithm of its ambiguity (the maximum number of strings that can be distorted into a given one). The other more restricted universal codes still work for large classes of natural channels. In a universal code, encoding is channel-independent, but the decoding function knows the type of channel. We allow the encoding and the decoding functions to share randomness, which is unavailable to the channel. There are two scenarios for the type of attack that a channel can perform. In the oblivious scenario, codewords belong to an additive group and the channel distorts a codeword by adding a vector from a fixed set. The selection is based on the message and the encoding function, but not on the codeword. In the Hamming scenario, the channel knows the codeword and is fully adversarial. For a universal code, there are two parameters of interest: the rate, which is the ratio between the message length k and the codeword length n, and the number of shared random bits. We show the existence in both scenarios of universal codes with rate 1-t/n - o(1), which is optimal modulo the o(1) term. The number of shared random bits is O(log n) in the oblivious scenario, and O(n) in the Hamming scenario, which, for typical values of the noise level, we show to be optimal, modulo the constant hidden in the O() notation. In both scenarios, the universal encoding is done in time polynomial in n, but the channel-dependent decoding procedures are in general not efficient. For some weaker classes of channels we construct universal codes with polynomial-time encoding and decoding., Comment: Removed the mentioning of online matching, which is not used here

It is known that the normalized algorithmic information distance $N$ is not computable and not semicomputable. We show that for all $\epsilon < 1/2$, there exist no semicomputable functions that differ from $N$ by at most~$\epsilon$. Moreover, for any computable function $f$ such that $|\lim_t f(x,y,t) - N(x,y)| \le \epsilon$ and for all $n$, there exist strings $x,y$ of length $n$ such that $\sum_t |f(x,y,t+1) - f(x,y,t)| \ge \Omega(\log n)$. This is optimal up to constant factors. We also show that the maximal number of oscillations of a limit approximation of $N$ is $\Omega(n/\log n)$. This strengthens the $\omega(1)$ lower bound from [K. Ambos-Spies, W. Merkle, and S.A. Terwijn, 2019, Normalized information distance and the oscillation hierarchy], see arXiv:1708.03583 ., Comment: 14 pages, 2 figures

In a lossless compression system with target lengths, a compressor ${\cal C}$ maps an integer $m$ and a binary string $x$ to an $m$-bit code $p$, and if $m$ is sufficiently large, a decompressor ${\cal D}$ reconstructs $x$ from $p$. We call a pair $(m,x)$ $\textit{achievable}$ for $({\cal C},{\cal D})$ if this reconstruction is successful. We introduce the notion of an optimal compressor ${\cal C}_\text{opt}$, by the following universality property: For any compressor-decompressor pair $({\cal C}, {\cal D})$, there exists a decompressor ${\cal D}'$ such that if $(m,x)$ is achievable for $({\cal C},{\cal D})$, then $(m+\Delta, x)$ is achievable for $({\cal C}_\text{opt}, {\cal D}')$, where $\Delta$ is some small value called the overhead. We show that there exists an optimal compressor that has only polylogarithmic overhead and works in probabilistic polynomial time. Differently said, for any pair $({\cal C}, {\cal D})$, no matter how slow ${\cal C}$ is, or even if ${\cal C}$ is non-computable, ${\cal C}_{\text{opt}}$ is a fixed compressor that in polynomial time produces codes almost as short as those of ${\cal C}$. The cost is that the corresponding decompressor is slower. We also show that each such optimal compressor can be used for distributed compression, in which case it can achieve optimal compression rates, as given in the Slepian-Wolf theorem, and even for the Kolmogorov complexity variant of this theorem. Moreover, the overhead is logarithmic in the number of sources, and unlike previous implementations of Slepian-Wolf coding, meaningful compression can still be achieved if the number of sources is much larger than the length of the compressed strings., Comment: 26 pages

We consider the notion of information distance between two objects x and y introduced by Bennett, G\'acs, Li, Vitanyi, and Zurek [1] as the minimal length of a program that computes x from y as well as computing y from x, and study different versions of this notion. It was claimed by Mahmud [11] that the prefix version of information distance equals max(K(x|y), K(y|) + O(1) (this equality with logarithmic precision was one of the main results of the paper by Bennett, G\'acs, Li, Vitanyi, and Zurek). We show that this claim is false, but does hold if the information distance is at least super logarithmic., Comment: Preliminary version, published for reference purposes

We give simplify the proofs of the 2 results in Marius Zimand's paper "Kolmogorov complexity version of Slepian-Wolf coding, proceedings of STOC 2017, p22--32". The first is a universal polynomial time compression algorithm: on input $\varepsilon > 0$, a number $k$ and a string $x$ it computes in polynomial time with probability $1-\varepsilon$ a program that outputs $x$ and has length $k + O(\log^2 (|x|/\varepsilon))$, provided that there exists such a program of length at most $k$. The second result, is a distributed compression algorithm, in which several parties each send some string to a common receiver. Marius Zimand proved a variant of the Slepian-Wolf theorem using Kolmogorov complexity (in stead of Shannon entropy). With our simpler proof we improve the parameters of Zimand's result.

The definition of conditional probability in case of continuous distributions was an important step in the development of mathematical theory of probabilities. How can we define this notion in algorithmic probability theory? In this survey we discuss the developments in this direction trying to explain what are the difficulties and what can be done to avoid them. Almost all the results discussed in this paper have been published (and we provide the references), but we tried to put them into perspective and to explain the proofs in a more intuitive way. We assume that the reader is familiar with basic notions of measure theory and algorithmic randomness.

We show that there exists a bitsequence that is not computably random for which its odd bits are computably random and its even bits are computably random relative to the odd bits. This implies that the uniform variant of van Lambalgen's theorem fails for computable randomness. (The other direction of van Lambalgen's theorem is known to hold in this case, and known to fail for the non-uniform variant.), Comment: 4 pages

Van Lambalgen's theorem states that a pair $(\alpha,\beta)$ of bitsequences is Martin-L\"of random if and only if $\alpha$ is Martin-L\"of random and $\beta$ is Martin-L\"of random relative to $\alpha$. In [Information and Computation 209.2 (2011): 183-197, Theorem 3.3], Hayato Takahashi generalized van Lambalgen's theorem for computable measures $P$ on a product of two Cantor spaces; he showed that the equivalence holds for each $\beta$ for which the conditional probability $P(\cdot | \beta)$ is computable. He asked whether this computability condition is necessary. We give a positive answer by providing a computable measure for which van Lambalgen's theorem fails. We also present a simple construction of a measure for which conditional measure is not computable. Such measures were first constructed by N. Ackerman, C. Freer and D. Roy in [Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 107-116. IEEE (2011)].

A sumtest for a discrete semimeasure $P$ is a function $f$ mapping bitstrings to non-negative rational numbers such that \[ \sum P(x)f(x) \le 1 \,. \] Sumtests are the discrete analogue of Martin-L\"of tests. The behavior of sumtests for computable $P$ seems well understood, but for some applications lower semicomputable $P$ seem more appropriate. In the case of tests for independence, it is natural to consider upper semicomputable tests (see [B.Bauwens and S.Terwijn, Theory of Computing Systems 48.2 (2011): 247-268]). In this paper, we characterize upper semicomputable sumtests relative to any lower semicomputable semimeasures using Kolmogorov complexity. It is studied to what extend such tests are pathological: can upper semicomputable sumtests for $m(x)$ be large? It is shown that the logarithm of such tests does not exceed $\log |x| + O(\log^{(2)} |x|)$ (where $|x|$ denotes the length of $x$ and $\log^{(2)} = \log\log$) and that this bound is tight, i.e. there is a test whose logarithm exceeds $\log |x| - O(\log^{(2)} |x|$) infinitely often. Finally, it is shown that for each such test $e$ the mutual information of a string with the Halting problem is at least $\log e(x)-O(1)$; thus $e$ can only be large for ``exotic'' strings., Comment: 10 pages

A $c$-short program for a string $x$ is a description of $x$ of length at most $C(x) + c$, where $C(x)$ is the Kolmogorov complexity of $x$. We show that there exists a randomized algorithm that constructs a list of $n$ elements that contains a $O(\log n)$-short program for $x$. We also show a polynomial-time randomized construction that achieves the same list size for $O(\log^2 n)$-short programs. These results beat the lower bounds shown by Bauwens et al. \cite{bmvz:c:shortlist} for deterministic constructions of such lists. We also prove tight lower bounds for the main parameters of our result. The constructions use only $O(\log n)$ ($O(\log^2 n)$ for the polynomial-time result) random bits . Thus using only few random bits it is possible to do tasks that cannot be done by any deterministic algorithm regardless of its running time.

In [3] a short proof is given that some strings have maximal plain Kolmogorov complexity but not maximal prefix-free complexity. The proof uses Levin's symmetry of information, Levin's formula relating plain and prefix complexity and Gacs' theorem that complexity of complexity given the string can be high. We argue that the proof technique and results mentioned above are useful to simplify existing proofs and to solve open questions. We present a short proof of Solovay's result [21] relating plain and prefix complexity: $K (x) = C (x) + CC (x) + O(CCC (x))$ and $C (x) = K (x) - KK (x) + O(KKK (x))$, (here $CC(x)$ denotes $C(C(x))$, etc.). We show that there exist $\omega$ such that $\liminf C(\omega_1\dots \omega_n) - C(n)$ is infinite and $\liminf K(\omega_1\dots \omega_n) - K(n)$ is finite, i.e. the infinitely often C-trivial reals are not the same as the infinitely often K-trivial reals (i.e. [1,Question 1]). Solovay showed that for infinitely many $x$ we have $|x| - C (x) \le O(1)$ and $|x| + K (|x|) - K (x) \ge \log^{(2)} |x| - O(\log^{(3)} |x|)$, (here $|x|$ denotes the length of $x$ and $\log^{(2)} = \log\log$, etc.). We show that this result holds for prefixes of some 2-random sequences. Finally, we generalize our proof technique and show that no monotone relation exists between expectation and probability bounded randomness deficiency (i.e. [6, Question 1])., Comment: 20 pages, 1 figure

Joseph Miller [16] and independently Andre Nies, Frank Stephan and Sebastiaan Terwijn [18] gave a complexity characterization of 2-random sequences in terms of plain Kolmogorov complexity C: they are sequences that have infinitely many initial segments with O(1)-maximal plain complexity (among the strings of the same length). Later Miller [17] showed that prefix complexity K can also be used in a similar way: a sequence is 2-random if and only if it has infinitely many initial segments with O(1)-maximal prefix complexity (which is n + K (n) for strings of length n). The known proofs of these results are quite involved; in this paper we provide simple direct proofs for both of them. In [16] Miller also gave a quantitative version of the first result: the 0'-randomness deficiency of a sequence {\omega} equals lim inf [n - C ({\omega}1 . . . {\omega}n)] + O(1). (Our simplified proof can also be used to prove this.) We show (and this seems to be a new result) that a similar quantitative result is also true for prefix complexity: 0'-randomness deficiency equals lim inf [n + K (n) -- K ({\omega}1 . . . {\omega}n)] + O(1)., Comment: 16 pages, 2 figures

Symmetry of information states that $C(x) + C(y|x) = C(x,y) + O(\log C(x))$. We show that a similar relation for online Kolmogorov complexity does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring $x_1x_2... x_n$ be the length of a shortest program that computes $x_2$ on input $x_1$, computes $x_4$ on input $x_1x_2x_3$, etc; and similar for odd complexity. We show that for all n there exist an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence $x_2x_1x_4x_3\ldots$, decreases the sum of odd and even complexity to $C(x)$., Comment: 20 pages, 7 figures

Sophistication and logical depth are two measures that express how complicated the structure in a string is. Sophistication is defined as the minimal complexity of a computable function that defines a two-part description for the string that is shortest within some precision; the second can be defined as the minimal computation time of a program that is shortest within some precision. We show that the Busy Beaver function of the sophistication of a string exceeds its logical depth with logarithmically bigger precision, and that logical depth exceeds the Busy Beaver function of sophistication with logarithmically bigger precision. We also show that this is not true if the precision is only increased by a constant (when the notions are defined with plain Kolmogorov complexity). Finally we show that sophistication is unstable in its precision: constant variations can change its value by a linear term in the length of the string., Comment: 9 pages

Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any standard Turing machine, it is possible to compute in polynomial time on input $x$ a list of polynomial size guaranteed to contain a O$(\log |x|)$-short program for $x$. We also show that there exists a computable function that maps every $x$ to a list of size $|x|^2$ containing a O$(1)$-short program for $x$. This is essentially optimal because we prove that for each such function there is a $c$ and infinitely many $x$ for which the list has size at least $c|x|^2$. Finally we show that for some standard machines, computable functions generating lists with $0$-short programs, must have infinitely often list sizes proportional to $2^{|x|}$.

Peter Gacs showed (Gacs 1974) that for every n there exists a bit string x of length n whose plain complexity C(x) has almost maximal conditional complexity relative to x, i.e., C(C(x)|x) > log n - log^(2) n - O(1). (Here log^(2) i = log log i.) Following Elena Kalinina (Kalinina 2011), we provide a simple game-based proof of this result; modifying her argument, we get a better (and tight) bound log n - O(1). We also show the same bound for prefix-free complexity. Robert Solovay showed (Solovay 1975) that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length): for some c there exist infinitely many x such that |x| - C(x) < c and |x| + K(|x|) - K(x) > log^(2) |x| - c log^(3) |x|. In fact, the results of Solovay and Gacs are closely related. Using the result above, we provide a short proof for Solovay's result. We also generalize it by showing that for some c and for all n there are strings x of length n with n - C (x) < c and n + K(n) - K(x) > K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. We also prove a close upper bound K(K(n)|n) + O(1). Finally, we provide a direct game proof for Joseph Miller's generalization (Miller 2006) of the same Solovay's theorem: if a co-enumerable set (a set with c.e. complement) contains for every length a string of this length, then it contains infinitely many strings x such that |x| + K(|x|) - K(x) > log^(2) |x| + O(log^(3) |x|)., Comment: 13 pages, 1 figure

We prove the formula C(a,b) = K(a|C(a,b)) + C(b|a,C(a,b)) + O(1) that expresses the plain complexity of a pair in terms of prefix and plain conditional complexities of its components., Comment: 4 pages

The m-sophistication of a finite binary string x is introduced as a generalization of some parameter in the proof that complexity of complexity is rare. A probabilistic near sufficient statistic of x is given which length is upper bounded by the m-sophistication of x within small additive terms. This shows that m-sophistication is lower bounded by coarse sophistication and upper bounded by sophistication within small additive terms. It is also shown that m-sophistication and coarse sophistication can not be approximated by an upper or lower semicomputable function, not even within very large error., Comment: 13 pages, draft

Ratios of universal enumerable semimeasures corresponding to hypotheses are investigated as a solution for statistical composite hypotheses testing if an unbounded amount of computation time can be assumed. Influence testing for discrete time series is defined using generalized structural equations. Several ideal tests are introduced, and it is argued that when Halting information is transmitted, in some cases, instantaneous cause and consequence can be inferred where this is not possible classically. The approach is contrasted with Bayesian definitions of influence, where it is left open whether all Bayesian causal associations of universal semimeasures are equal within a constant. Finally the approach is also contrasted with existing engineering procedures for influence and theoretical definitions of causation., Comment: 29 pages, 3 figures, draft

It is shown that the length of the algorithmic minimal sufficient statistic of a binary string x, either in a representation of a finite set, computable semimeasure, or a computable function, has a length larger than the computational depth of x, and can solve the Halting problem for all programs with length shorter than the m-depth of x. It is also shown that there are strings for which the algorithmic minimal sufficient statistics can contain a substantial amount of information that is not Halting information. The weak sufficient statistic is introduced, and it is shown that a minimal weak sufficient statistic for x is equivalent to a minimal typical model of x, and to the Halting problem for all strings shorter than the BB-depth of x., Comment: 25 pages

We show that there are infinitely many binary strings z, such that the sum of the on-line decision complexity of predicting the even bits of z given the previous uneven bits, and the decision complexity of predicting the uneven bits given the previous event bits, exceeds the Kolmogorov complexity of z by a linear term in the length of z., Comment: 10 pages