Interval Selection in Sliding Windows

TL;DR

The study of the Interval Selection problem in the (streaming) sliding window model of computation is initiated, with an improvement over the smooth histogram technique, which constrains the shape of an optimal solution and allows us to target optimal intervals differently.

Abstract

We initiate the study of the Interval Selection problem in the (streaming) sliding window model of computation. In this problem, an algorithm receives a potentially infinite stream of intervals on the line, and the objective is to maintain at every moment an approximation to a largest possible subset of disjoint intervals among the $L$ most recent intervals, for some integer $L$. We give the following results: - In the unit-length intervals case, we give a $2$-approximation sliding window algorithm with space $\tilde{\mathrm{O}}(|OPT|)$, and we show that any sliding window algorithm that computes a $(2-\varepsilon)$-approximation requires space $Ω(L)$, for any $\varepsilon > 0$. - In the arbitrary-length case, we give a $(\frac{11}{3}+\varepsilon)$-approximation sliding window algorithm with space $\tilde{\mathrm{O}}(|OPT|)$, for any constant $\varepsilon > 0$, which constitutes our main result. We also show that space $Ω(L)$ is needed for algorithms that compute a $(2.5-\varepsilon)$-approximation, for any $\varepsilon > 0$. Our main technical contribution is an improvement over the smooth histogram technique, which consists of running independent copies of a traditional streaming algorithm with different start times. By employing the one-pass $2$-approximation streaming algorithm by Cabello and Pérez-Lantero [Theor. Comput. Sci. '17] for Interval Selection on arbitrary-length intervals as the underlying algorithm, the smooth histogram technique immediately yields a $(4+\varepsilon)$-approximation in this setting. Our improvement is obtained by forwarding the structure of the intervals identified in a run to the subsequent run, which constrains the shape of an optimal solution and allows us to target optimal intervals differently.

Figures (6)

• Figure 1: Approximation factors achievable in the streaming and sliding window models. All algorithms use space $\tilde{\mathrm{O}}(|OPT|)$, while all lower bound results are to be interpreted in that achieving the stated approximation guarantee requires space $\Omega(n)$ (streaming) or $\Omega(L)$ (sliding window model). The lower bound results hold for any $\varepsilon > 0$.
• Figure 2: This figure illustrates the instances created by Alice and Bob in the proof of Theorem \ref{['thm:lb-unit-length']} for an instance of $\textsf{Index}_{L-2}$ with $X[J] = 1$. The dashed intervals on the upper part correspond to the zero elements of the bitvector $X$. The red intervals $I_1$, $I_2$ correspond to expired intervals. $I_J$ is the only non-expired interval disjoint with the special interval $I_{L-1}$. Since $X[J] = 1$, the optimal solution is of size $2$. If $X[J]$ was equal to $0$ then the interval $I_J$ would not be disjoint with $I_{L-1}$, and, thus, an optimal solution would be of size $1$.
• Figure 3: Key Properties of the $\mathcal{CP}$ Algorithm.
• Figure 4: Key Properties of the Smooth Histogram Technique.
• Figure 5: This figure illustrates the intervals created by Alice, Bob and Charlie in the proof of Theorem \ref{['thm:lb-arbitrary-length']} for an instance of $\textsf{Chain}_3(n)$ where $n = \frac{L-2}{3}$ with $X_1[J_1] = X_2[J_2] = 1$. The red intervals in the figure ($I_1(1),I_1(J_1 - 1)$,$I_2(1)$,$I_2(J_1 - 1)$) correspond to expired intervals. The optimal solution is $\{I_1(J_1), I_2(J_1), I_{J_2}, I'_{J_2}, I_3(J_2)\}$ of size 5 . Otherwise, if $X_1[J_1] = X_2[J_2] = 0$, then the optimal solution would have been of size 2. All intervals $I_3(i)$ are disjoint from $I_1(J_1)$ and $I_2(J_2)$. However, they intersect with $I_1(J_1 + 1)$ and $I_2(J_2 + 1)$ as emphasized by the vertical dashed lines. Intervals $I_3(i)$ for $n+2(J_1 - 1) \ge i > n$ have been omitted as they do not impact the optimal solution and their only role is to advance the sliding window.

Theorems & Definitions (30)

• Theorem 1: e.g. Jayram2008TheOC
• Theorem 2: sundaresan2024optimal
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 1
• proof
• Theorem 5
• proof