# Ubiquitous Integrity via Network Integration and Parallelism—Sustaining Pedestrian/Bike Urbanism

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Mathematical Preliminaries

_{0}, x

_{1}, x

_{2},…x

_{n−}

_{1}). A path is called a Hamiltonian path if its nodes are distinct and span V. A cycle is a path of at least three nodes such that the first node is the same as the last node. A cycle is called a Hamiltonian cycle or Hamiltonian if its nodes are distinct except for the first node and the last node, and if they span V [10].

_{p}-Hamiltonian. A graph G is 1-edge Hamiltonian if G-e is Hamiltonian for any e ∈ E; moreover, if there is a Hamiltonian path between any pair of nodes {c, d} with c ∈ A and d ∈ B, then the bipartite graph G is Hamiltonian laceable.

**mod**n); (2) j = l and k = i − 1 if i + j is even; and (3) i = 0, k = m − 1, and l =j + d(

**mod**n) if j is even.

**mod**m) if i + j is odd or j = n − 1; and (3) j = l, k = i − 1(

**mod**m) if i + j is even or j = 0. The configuration of SW(m, n) is shown in Figure 1(a,d).

_{p}-Hamiltonian if n ≥ 6 or m = 2, n ≥ 4 [13]. When m and n are positive integers with n, m − n/2 being even, GHT(m, n, n/2) is proved 1-edge Hamiltonian if n ≥ 4; 1

_{p}-Hamiltonian if n ≥ 6 or m = 2, n ≥ 4 [14]. Besides, SW(m, n) is proved 1-edge Hamiltonian, 1

_{p}-Hamiltonian [15]. Thus, the fault-tolerance in which we are engaged is systematically based. Moreover, GHT(m, n, 0), GHT(m, n, n/2), and SW(m, n) are Hamiltonian laceable if m, n ≥ 4 integers [16,17].

_{1}= (u

_{1}, u

_{2}, …, u

_{n}

_{(G)}) and P

_{2}= (v

_{1}, v

_{2}, …, v

_{n}

_{(G)}) of G from u to v are independent if u = u

_{1}= v

_{1}, v = u

_{n}

_{(G)}= v

_{n}

_{(G)}, and u

_{i}≠ v

_{i}, for every 1 < i < n (G). A set of Hamiltonian paths, {P

_{1}, P

_{2}, …, P

_{k}}, of G from u to v is mutually independent if any two different Hamiltonian paths are independent from u to v. The mechanism of mutually independent Hamiltonian paths (MIHP) can be applied to parallel processing [18]. Such a feature is also considered for secret communications [19,20]. It is proved that SW(m, n) has MIHP performance [8]; and it is believed that GHT(m, n, 0) and GHT(m, n, n/2) can have such performance if nodes are enough (e.g., more than 16).

**Figure 1.**Link adaptability in wide area communication network, (

**a**) urban dedicated short range communication network with Spider-Web networks’ integration and fractal connectivity performances; (

**b**) wireless communication quality affected by link adaptability; (

**c**) cellular communication modeled by honeycomb tori, generalized honeycomb torus (GHT) networks; (

**d**) cellular communication modeled by Spider-Web networks.

_{n}

_{−}

_{1}… b

_{i}… b

_{0}be an n-bit binary string. For any j, 0 ≤ j ≤ n − 1, we use (u)

^{j}to denote the binary string b

_{n}

_{−}

_{1}…b

_{j}… b

_{0}. Moreover, we use (u)

_{j}to denote the bit b

_{j}of u. The Hamming weight of u, denoted by wH(u), is the value of |{0 ≤ i ≤ n − 1|(u)

_{i}= 1}|. The n-cube (or hypercube) Q

_{n}consists of 2

^{n}nodes and n2

^{n}

^{−1}links. Each node corresponds to an n-bit binary string. Two nodes, u and v, are adjacent if and only if v = (u)

^{j}for some j, and we call link (u, (u)

^{j}) j-dimensional. The Hamming distance between u and v, denoted by h(u, v), is defined to be the number of elements in {0 ≤ i ≤ n − 1|(u)

_{i}≠ (v)

_{i}}. Hence, two nodes, u and v, are adjacent if and only if h(u, v) = 1.

_{n}has n2

^{n}nodes, labeled as (l, x), where l is an integer between 0 and n − 1, and x is an n-bit and x is the processor node with an n-bit binary string. Two vertices (l, x) and (l’, y) are adjacent if and only if x = y and |l − l’| = 1 or l = l’ and y = (x)

^{l}. In the latter case, x and y only differ in the position l. The edges that connect (l, x) to its neighbors (l + 1, x) and (l − 1, x) are called cycle-edges. Moreover these cycle-edges form a cycle of length n called a fundamental cycle defined by x (Figure 2), which can represent a node composed of ring networked processors. L(n) is the set offering all possible lengths of cycles in CCC

_{n}[9]; for n = 2, CCC

_{n}is just the cycle of length 8. The possible-length parameter can be applied for systematic checking and design on affiliated processors.

**Figure 2.**Relationship between Hypercubes and Cube-Connected-Cycles. (

**a**) CCC

_{2}; (

**b**) CCC

_{3}; (

**c**) CCC

_{4}and ring networked module.

## 4. Proof Examples (two MIHP on GHT(m, n, n/2) )

#### 4.1. m Even, n ≥ 4; y1 = y2, Even; (x2 − x1) Odd

**Figure 3.**Mutually independent Hamiltonian paths (MIHP) performance exist in GHT(2,4,2) (

**a**) without vertical extension (X-ext.); (

**b**) with vertical extension (X-ext.).

#### 4.2. m Even, n ≥ 4; y1 = y2, Odd; (x2 − x1) Odd

## 5. Network Proposal

#### 5.1. Along Streets—A Pedestrian/Bike DSRC Proposal

#### 5.2. Probe Proposal

^{n}sup-nodes, or n2

^{n}nodes totally in a planned zone. A probe integrates all its nodes’ operation, and then wirelessly transmits information to other probes, and other areas via cellular communication and global positioning systems.

#### 5.3. Linking Alternatives

- (1)
- (2)
- On cellular communication, both SW(6, n) and GHT(m, 6m, 3m) are proposed (Figure 1(c,d)) with MIHP, fault-tolerance performances.
- (3)
- Radial-ring, SW networks, have more fault-tolerance than ring networks. The area DSRC network can be configured as an integral SW network. Naturally, radial-ring can be grouped, and even configured as a radial-ring of radial-rings (Figure 1(a)).

## 6. Conclusions

## Acknowledgment

## Conflict of Interest

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**MDPI and ACS Style**

Hsu, L.-Y.
Ubiquitous Integrity via Network Integration and Parallelism—Sustaining Pedestrian/Bike Urbanism. *Algorithms* **2013**, *6*, 459-470.
https://doi.org/10.3390/a6030459

**AMA Style**

Hsu L-Y.
Ubiquitous Integrity via Network Integration and Parallelism—Sustaining Pedestrian/Bike Urbanism. *Algorithms*. 2013; 6(3):459-470.
https://doi.org/10.3390/a6030459

**Chicago/Turabian Style**

Hsu, Li-Yen.
2013. "Ubiquitous Integrity via Network Integration and Parallelism—Sustaining Pedestrian/Bike Urbanism" *Algorithms* 6, no. 3: 459-470.
https://doi.org/10.3390/a6030459