Breaking the Barrier of 2 for the Competitiveness of Longest Queue Drop

TL;DR

The paper resolves a long-standing open problem by proving that the Longest Queue Drop (LQD) algorithm for buffer management in shared-memory switches is $1.6918$-competitive, improving the previous $2$-competitive upper bound. It introduces a phase-based profit-splitting framework that distributes LQD’s profit across queues and bounds the offline optimum via a careful mapping of OPT-extra packets to LQD activity. A novel potential function and a decomposition into L-increase and S-increase enable tight per-queue analyses, incorporating live/dying queue dynamics and a mapping scheme that cancels portions of OPT’s advantage. The result has practical impact on online buffer management by providing the first constant-factor improvement for LQD without restrictions on the number of ports or buffer size, and the techniques may inform broader online-algorithm analyses in networking and queueing systems.

Abstract

We consider the problem of managing the buffer of a shared-memory switch that transmits packets of unit value. A shared-memory switch consists of an input port, a number of output ports, and a buffer with a specific capacity. In each time step, an arbitrary number of packets arrive at the input port, each packet designated for one output port. Each packet is added to the queue of the respective output port. If the total number of packets exceeds the capacity of the buffer, some packets have to be irrevocably evicted. At the end of each time step, each output port transmits a packet in its queue and the goal is to maximize the number of transmitted packets. The Longest Queue Drop (LQD) online algorithm accepts any arriving packet to the buffer. However, if this results in the buffer exceeding its memory capacity, then LQD drops a packet from whichever queue is currently the longest, breaking ties arbitrarily. The LQD algorithm was first introduced in 1991, and is known to be $2$-competitive since 2001. Although LQD remains the best known online algorithm for the problem and is of practical interest, determining its true competitiveness is a long-standing open problem. We show that LQD is 1.6918-competitive, establishing the first $(2-\varepsilon)$ upper bound for the competitive ratio of LQD, for a constant $\varepsilon>0$.

Figures (4)

• Figure 1: An example of the buffer configuration for LQD and OPT during some time step $t$ while processing the incoming packets, for buffer of size $M=65$. The blue, north-west shaded areas (aligned to the left) correspond to the packets in queues of LQD and the red, north-east shaded areas (aligned to the right) to the queues of OPT. For instance, we have $s_{\textsf{OPT}\xspace}^t(6)= 14$, and $s_{\textsf{LQD}\xspace}^t(6) = 7$. Furthermore, $s_{\text{max}}^t = 12$ is the maximal size of a queue for LQD. Note that an OPT-extra packet is going to be transmitted from queue $9$ in step $t$, and as queue $9$ is empty for LQD, no further packet arrives to this queue by assumption (A1). All the queues with an index $\ge 10$ are inactive (i.e., empty in both buffers). According to Definition \ref{['def:overflow']}, queues $1, 2,$ and $3$ overflow. As an example, assume that further $3$ packets arrive into queue $8$ and the LQD buffer is already full. Then LQD would first evict a packet from queue $1$ and then select two of the queues $1$, $2$ or $3$, dropping one packet from each selected queue.
• Figure 2: An example of a life-cycle of a queue $q$ that overflows in some steps, with $t_{q}$ being the last such step; the queue is depicted similarly as in Fig. \ref{['fig:buffer']}. Note that $e_{q} = 5$. We remark that in "hard instances", OPT would keep just one packet in $q$ before time $t_{q}$, while the size of the queue in the LQD buffer varies. Thus, the queue takes almost no space in the OPT buffer, while it is larger in the LQD buffer and both OPT and LQD gain packets from this queue in every step before $t_{q}$. At $t_{q}$, however, the situation reverses: OPT stores many more packets than LQD in the queue, and if no packets arrive to this queue after $t_{q}$, OPT gains a number of OPT-extra packets. Note that $e_{q}$ is upper-bounded by the number of additional packets OPT stores in $q$ compared to LQD.
• Figure 3: Intuition for splitting the LQD profit in some step $\tau_i$; the queues are depicted similarly as in Fig. \ref{['fig:buffer']}. The queues $q$ on the left, in which OPT stores just one packet, typically satisfy $\tau_i < t_q$, i.e., they overflow after $\tau_i$. On the other hand, queues on the right, where OPT stores more packets than LQD, do not overflow after $\tau_i$ and OPT transmits some OPT-extra packets from them. We note that in hard instances, the maximum size of a queue for LQD, denoted $s^t_{\max}$, may fluctuate over time, as witnessed in the lower bound of $\approx 1.44$ in bochkov2019new.
• Figure 4: An example of the LQD buffer in step $\tau' = \tau_{i+1}$ for illustrating the proof of Lemma \ref{['lem:dying_bound']}. Note that $s^{i+1}_{\max} = 10$ and that queues $5-9$ overflow (and thus the LQD buffer is full). However, only queue 5 becomes dying as it overflows for the last time at $\tau'$, i.e., $t_5 = \tau'$. Moreover, queue 12 was empty in step $\tau_i$ and queue 1 will become empty for LQD just after packets are transmitted in step $\tau'$ (note that no further packets will arrive to queue 1 after step $\tau'$ by assumption (A1)). We have that $\mathcal{L}' = \{5, 6, \dots, 12\}$. Finally, $\sigma^{i+1}_q = 56/7 = 8$, since there are 56 packets in 7 live queues $\mathcal{L}^{i+1}_q$.

Theorems & Definitions (29)

• Theorem 1
• Definition 2
• Definition 4
• proof
• proof
• Lemma 9
• proof
• Lemma 10
• proof
• Lemma 11