Low (complexity) | EPFL Graph Search

In computational complexity theory, a language B (or a complexity class B) is said to be low for a complexity class A (with some reasonable relativized version of A) if AB = A; that is, A with an oracle for B is equal to A. Such a statement implies that an abstract machine which solves problems in A achieves no additional power if it is given the ability to solve problems in B at unit cost. In particular, this means that if B is low for A then B is contained in A. Informally, lowness means that problems in B are not only solvable by machines which can solve problems in A, but are “easy to solve”. An A machine can simulate many oracle queries to B without exceeding its resource bounds. Results and relationships that establish one class as low for another are often called lowness results. The set of languages low for a complexity class A is denoted Low(A). Classes that are low for themselves Several natural complexity classes are known to be low for themselves. Such a clas

Coding schemes for broadcast erasure channels with feedback: the two multicast case

Christina Fragouli, Efe Onaran

We consider the single hop broadcast packet erasure channel (BPEC) with two multicast sessions (each of them destined to a different group of

N

users) and regularly available instantaneous receiver ACK/NACK feedback. Using the insight gained from recent work on BPEC with unicast and degraded messages [1], [2], we propose a virtual queue based session-mixing algorithm, which does not rely on knowledge of channel statistics and achieves capacity for

N=2

and iid erasures. Since the optimal extension of this algorithm to

N>2

is not straightforward, we then describe a low complexity algorithm which outperforms standard timesharing for arbitrary

N

and is actually asymptotically better than timesharing, for any finite

N

, as the erasure probability goes to zero. We finally provide, through an information-theoretic analysis, sufficient but not necessary asymptotic conditions between

N

and

n

(the number of transmissions) for which the achieved sum rate, under \textit{any} coding scheme, is essentially identical to that of timesharing.

2013

Routing and search on large scale networks

Dominique Florian Tschopp

In this thesis, we address two seemingly unrelated problems, namely routing in large wireless ad hoc networks and comparison based search in image databases. However, the underlying problem is in essence similar and we can use the same strategy to attack those two problems. In both cases, the intrinsic complexity of the problem is in some sense low, and we can exploit this fact to design efficient algorithms. A wireless ad hoc network is a communication network consisting of wireless devices such as for instance laptops or cell phones. The network does not have any fixed infrastructure, and hence nodes which cannot communicate directly over the wireless medium must use intermediate nodes as relays. This immediately raises the question of how to select the relay nodes. Ideally, one would like to find a path from the source to the destination which is as short as possible. The length of the found path, also called route, typically depends on how much signaling traffic is generated in order to establish the route. This is the fundamental trade-off that we will investigate in this thesis. As mentioned above, we try and exploit the fact that the communication network is intrinsically low-dimensional, or in other words has low complexity. We show that this is indeed the case for a large class of models and that we can design efficient algorithms for routing that use this property. Low dimensionality implies that we can well embed the network in a low-dimensional space, or build simple hierarchical decompositions of the network. We use both those techniques to design routing algorithms. Comparison based search in image databases is a new problem that can be defined as follows. Given a large database of images, can a human user retrieve an image which he has in mind, or at least an image similar to that image, without going sequentially through all images? More precisely, we ask whether we can search a database of images only by making comparisons between images. As a case in point, we ask whether we can find a query image q only by asking questions of the type "does image q look more like image A or image B"? The analogous to signaling traffic for wireless networks would here be the questions we can ask human users in a learning phase anterior to the search. In other words, we would like to ask as few questions as possible to pre-process and prepare the database, while guaranteeing a certain quality of the results obtained in the search phase. As the underlying image space is not necessarily metric, this raises new questions on how to search spaces for which only rank information can be obtained. The rank of A with respect to B is k, if A is B's kth nearest neighbor. In this setup, low-dimensionality is analogous to the homogeneity of the image space. As we will see, the homogeneity can be captured by properties of the rank relationships. In turn, homogeneous spaces can be well decomposed hierarchically using comparisons. Further, it allows us to design good hash functions. To design efficient algorithms for these two problems, we can apply the same techniques mutatis mutandis. In both cases, we relied on the intuition that the problem has a low intrinsic complexity, and that we can exploit this fact. Our results come in the form of simulation results and asymptotic bounds.

EPFL2010

Fast Low-Complexity Decoders for Low-Rate Polar Codes

Alexios Konstantinos Balatsoukas Stimming, Pascal Giard

Polar codes are capacity-achieving error-correcting codes with an explicit construction that can be decoded with low-complexity algorithms. In this work, we show how the state-of-the-art low-complexity decoding algorithm can be improved to better accommodate low-rate codes. More constituent codes are recognized in the updated algorithm and dedicated hardware is added to efficiently decode these new constituent codes. We also alter the polar code construction to further decrease the latency and increase the throughput with little to no noticeable effect on error-correction performance. Rate-flexible decoders for polar codes of length 1024 and 2048 are implemented on FPGA. Over the previous work, they are shown to have from 22% to 28% lower latency and 26% to 34% greater throughput when decoding low-rate codes. On 65 nm ASIC CMOS technology, the proposed decoder for a (1024, 512) polar code is shown to compare favorably against the state-of-the-art ASIC decoders. With a clock frequency of 400 MHz and a supply voltage of 0.8 V, it has a latency of 0.41 μs and an area efficiency of 1.8 Gbps/mm2 for an energy efficiency of 77 pJ/info. bit. At 600 MHz with a supply of 1 V, the latency is reduced to 0.27 μs and the area efficiency increased to 2.7 Gbps/mm2 at 115 pJ/info. bit.

2018