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- W4229076247 abstract "Simple tabulation hashing dates back to Zobrist in 1970 and is defined as follows: Each key is viewed as $c$ characters from some alphabet $Sigma$, we have $c$ fully random hash functions $h_0, ldots, h_{c - 1} colon Sigma to {0, ldots, 2^l - 1}$, and a key $x = (x_0, ldots, x_{c - 1})$ is hashed to $h(x) = h_0(x_0) oplus ldots oplus h_{c - 1}(x_{c - 1})$ where $oplus$ is the bitwise XOR operation. The previous results on tabulation hashing by P{v a}tra{c s}cu and Thorup~[J.ACM'11] and by Aamand et al.~[STOC'20] focused on proving Chernoff-style tail bounds on hash-based sums, e.g., the number keys hashing to a given value, for simple tabulation hashing, but their bounds do not cover the entire tail. Chaoses are random variables of the form $sum a_{i_0, ldots, i_{c - 1}} X_{i_0} cdot ldots cdot X_{i_{c - 1}}$ where $X_i$ are independent random variables. Chaoses are a well-studied concept from probability theory, and tight analysis has been proven in several instances, e.g., when the independent random variables are standard Gaussian variables and when the independent random variables have logarithmically convex tails. We notice that hash-based sums of simple tabulation hashing can be seen as a sum of chaoses that are not independent. This motivates us to use techniques from the theory of chaoses to analyze hash-based sums of simple tabulation hashing. In this paper, we obtain bounds for all the moments of hash-based sums for simple tabulation hashing which are tight up to constants depending only on $c$. In contrast with the previous attempts, our approach will mostly be analytical and does not employ intricate combinatorial arguments. The improved analysis of simple tabulation hashing allows us to obtain bounds for the moments of hash-based sums for the mixed tabulation hashing introduced by Dahlgaard et al.~[FOCS'15]." @default.
- W4229076247 created "2022-05-08" @default.
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- W4229076247 date "2022-05-03" @default.
- W4229076247 modified "2023-09-23" @default.
- W4229076247 title "Understanding the Moments of Tabulation Hashing via Chaoses" @default.
- W4229076247 doi "https://doi.org/10.48550/arxiv.2205.01453" @default.
- W4229076247 hasPublicationYear "2022" @default.
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