Load thresholds for cuckoo hashing with double hashing

In k-ary cuckoo hashing, each of cn objects is associated with k random buckets in a hash table of size n. An l-orientation is an assignment of objects to associated buckets such that each bucket receives at most l objects. Several works have determined load thresholds c^* = c^*(k,l) for k-ary cuckoo hashing; that is, for c c^* no l-orientation exists with high probability. A natural variant of k-ary cuckoo hashing utilizes double hashing, where, when the buckets are numbered 0,1,...,n-1, the k choices of random buckets form an arithmetic progression modulo n. Double hashing simplifies implementation and requires less randomness, and it has been shown that double hashing has the same behavior as fully random hashing in several other data structures that similarly use multiple hashes for each object. Interestingly, previous work has come close to but has not fully shown that the load threshold for k-ary cuckoo hashing is the same when using double hashing as when using fully random hashing. Specifically, previous work has shown that the thresholds for both settings coincide, except that for double hashing it was possible that o(n) objects would have been left unplaced. Here we close this open question by showing the thresholds are indeed the same, by providing a combinatorial argument that reconciles this stubborn difference

Power Allocation and Cooperative Diversity in Two-Way Non-Regenerative Cognitive Radio Networks

In this paper, we investigate the performance of a dual-hop block fading cognitive radio network with underlay spectrum sharing over independent but not necessarily identically distributed (i.n.i.d.) Nakagami-$m$ fading channels. The primary network consists of a source and a destination. Depending on whether the secondary network which consists of two source nodes have a single relay for cooperation or multiple relays thereby employs opportunistic relay selection for cooperation and whether the two source nodes suffer from the primary users' (PU) interference, two cases are considered in this paper, which are referred to as Scenario (a) and Scenario (b), respectively. For the considered underlay spectrum sharing, the transmit power constraint of the proposed system is adjusted by interference limit on the primary network and the interference imposed by primary user (PU). The developed new analysis obtains new analytical results for the outage capacity (OC) and average symbol error probability (ASEP). In particular, for Scenario (a), tight lower bounds on the OC and ASEP of the secondary network are derived in closed-form. In addition, a closed from expression for the end-to-end OC of Scenario (a) is achieved. With regards to Scenario (b), a tight lower bound on the OC of the secondary network is derived in closed-form. All analytical results are corroborated using Monte Carlo simulation method