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Signal Filtering to Obtain Number of Hamiltonian Paths 1:3

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Signal Filtering to Obtain Number of Hamiltonian Paths 1:3 1 Signal filtering to obtain number of Hamiltonian paths BRYCE KIM This paper consists of two parts. First, the (undirected) Hamiltonian path problem is reduced to a signal filtering problem - number of Hamiltonian paths becomes amplitude at zero frequency for (a combination of) sinusoidal signal f(t) that encodes a graph. Then a ’divide and conquer’ strategy to filtering out wide bandwidth components of a signal is suggested - one filters out angular frequency 1/2 to 1, then 1/4 to 1/2, then 1/8 to 1/4 and so on. An actual implementation of this strategy involves careful local polynomial extrapolation using numerical differentiation filters. When conjectures regarding required number of samples for specified filter designs and time complexity of obtaining filter coefficients hold, P=NP conditionally. CCS Concepts: • Mathematics of computing → Paths and connectivity problems; • Theory of com- putation → Complexity classes; • Hardware → Digital signal processing. Additional Key Words and Phrases: digital signal processing, digital filter, numerical differentiation, extrapo- lation, lowpass wide bandwidth filtering, Hamiltonian path problem, P=NP 1 INTRODUCTION This paper consists of two parts (1) and (2): (1) function encoding of an undirected graph into 5 (C) such that the zero-frequency amplitude 0 of 5 (C) is the number of Hamiltonian paths: =ℎ. Furthermore, Algorithm 1 computes G (C) in polynomial time relative to |+ | ≡ [ for each C. (2) lowpass wide bandwidth filtering strategy, consisting of two sub-strategies: frequency bi- nary divide-and-conquer and numerical differentiation filter-based local polynomial extrap- olation sub-strategies. (3) Filtering out 5 (C) to obtain its zero frequency amplitude takes polynomial time relative [, when conjectures of Equation (2), Equation (3) and Equation (6), stating the minimal required filter order bound, the filter coefficient magnitude upper bound and the filter coefficient time complexity bound, hold. This part dominates time complexity, which one can state as : $ ([ 552 ), with :5 5 2 defined in Equation (7). • Corresponding to above (1): a way of encoding an undirected graph = (+,) (with |+ | = [) into a combination 5 (C) of sinusoidal signals, with 5 (C) ∈ C and C ∈ R, is shown. In frequency arXiv:1504.05429v11 [cs.OH] 4 Apr 2021 domain (l) of 5 (C), (0) turns out to be the number of Hamiltonian paths. Therefore, find- ing the number of Hamiltonian paths is reduced to finding (0) by lowpass filtering. This is done by assigning each vertex (with assigned ordinal 8 when vertices are ordered 8 from 1 to [) angular frequency of E8 = [ appropriately. Let a sum of vertices refers to a sum of these vertex angular frequencies. Then each [-walk F - walk of length [ − 1 allowed in - is assigned angular frequency lF that is the sum of all vertices visited. For example, if E1 is visited twice and E2 is visited once in 3-walk F, then lF = 2E1 +E2. The goal then is to assign 8lFC 8lFC F with signal 4 , and then sum up all possible [-walk signals to form G (C) = F 4 . The angular frequency of Hamiltonian paths in G (C) then would be [ℎ = 8 E8. To make Í signal filtering convenient, we shift the Hamiltonian path angular frequency from [ℎ to zero by ~(C) = G (C)4−8[ℎC . Then we re-scale time by 5 (C) = ~(C/[[+1). Í The remaining issue then is to demonstrate that the Hamiltonian path angular frequency is not shared by other paths - this is done by utilizing the basis representation theorem, with Author’s address: Bryce Kim, [email protected]. 1:2 Bryce Kim basis being |+ | ≡ [ and vertex frequencies assigned as + = {[,[2,..,[[ }, given that each [-walk (a walk of length [ − 1) cannot visit the same vertex [ times. The rest of the proof that Algorithm 1, which computes G (C), works as advertised is given in subsection 2.4. • Corresponding to above (2): the problem with 5 (C) turns out to be that non-zero angular fre- quencies of 5 (C) with potential non-zero amplitude have domain of 1/[[+1 ≤ |l| ≤ 1. That is, the least positive angular frequency with potential non-zero amplitude is 1/[[+1, with maximum angular frequency with potential non-zero amplitude being 1. Conventionally, this filtering problem is considered to be infeasible, requiring exponential computational re- source (relative to [). This paper presents a ‘divide and conquer’ strategy that demonstrates : polynomial computational resource, in sense that time complexity is $ ([ 552 ), where :5 5 2 is some constant, assuming a particular digital filter order upper bound when given filter de- sign constraints and polynomial time complexity relative to [ in obtaining filter coefficients. In that sense % = #% up the filter order bound and coefficient time complexity assumptions. This is done by effectively first filtering out 1/2 of active angular frequencies, extrapolate new samples from filter output samples, and then filtering out 1/4 of originally active an- gular frequencies, extrapolate new samples and then filtering out 1/8 of originally active angular frequencies and so forth. This allows us the continued use of the same modest cut- off frequency, instead of an extremely low cutoff frequency that 5 (C) seems to demand for. Extrapolation error is controlled by re-doing numerical differentiation whenever new sam- ples are extrapolated using a constructed local polynomial that utilize numerical differenti- ation data - this allows us to treat extrapolation errors as if they come from a combination of sinusoid inputs, given that numerical differentiation filters admit the frequency response interpretation. This allows us to tame down extrapolation errors. Error analysis then is con- ducted with heavy use of Parseval’s theorem and by the ‘extrapolation error of extrapolation error’ strategy. 1.1 Preliminary assumption Sufficiently large |+ | ≡ [ > [0 would be assumed throughout the paper for simplification purposes. 1.2 Notation style G G For notation simplicity, it would be assumed that [[ ≡ [ ([ ) . 9, in contrast to typical engineering convention, would not refer to an imaginary number. Depending on contexts, 8 would either be used as an imaginary number or index 8, as typical in mathematical convention. : would refer to 2 2 2 positive natural number constants, with different subscripts. For notation, 01 ≡ 0 (1 ) ≠ 01 . By cycles, they would not refer to graph-theoretic cycles or Hamiltonian cycles and would be defined differently. For formula that are technically not equations, they may still be referred to as equations for expositional convenience. Superscripts always represent exponentiation. Sampling interval is al- ways assumed to be ΔC = 1. I ≡ #5 + #3 + 1, I2 ≡ 2(#5 + #3 )+ #5 − 2. Time refers to variable C in functions such as 5 (C). Signal filtering to obtain number of Hamiltonian paths 1:3 1.3 Preliminary terminology A ‘sinusoid(al) contribution’ would always refer to a combination of sinusoid(al) contributions. That is, (a combination of) sinusoid(al) contributions, where the term in the parenthesis would often be not written. From here on, each vertex would be labelled with its angular frequency, instead of its ordinal based on order from 1 to [, as + = {[,[2,..,[[ } shows. Each vertex with assigned ordinal 8 is assigned vertex label and angular frequency of [8. A sum of vertices then refers to a sum of vertex angular frequencies (labels). In contrast to vertices, a sum @(C) of walks refers to a sum of signals with walk angular frequen- 8lF,@ cies. Or formally notated, @(C) = F 4 C, where F refers to each walk being summed up to form @(C), and lF,@ refers to angular frequency of walk F. A [-walk refer to a walk of lengthÍ [ − 1, with restrictions that edges connecting one vertex to itself are disallowed and that a walk has to respect allowed edge connections. Edges may be used more than once. Length of a walk refers to the number of times any edge is walked upon by the walk. Or equivalently, it refers to the number of times any vertex is visited minus one. Each walk can visit one vertex more than once, as far as edge connections allow it. 1.4 Style issues in exponents and subscripts Due to the style of the manuscript, exponents and superscripts may not be clearly identified. \2 in 6\2 (C), F\2,9 (C), W\2 (C), k\2 (C), W\2,9 (C), k\2,9 (C), h\2,9 (C) and W\2,9 (C) of subsubsection 3.12.3 are 2 :105 subscripts. The denominator of : should be read as two to the power of [ . Similarly, de- 2[ 105 : .. [ 333.. 2[ :333.. nominator 2 in : should be read as two to the power of [ , where 333.. refers to 2[ 333.. : some random subscript. 2[ 333.. must be distinguished with 2[.., which refers to 2 times [... 2 FUNCTION ENCODING OF GRAPH The idea is to encode or translate an undirected graph as a computation circuit - for example, for graph in Figure 1, the circuit of Figure 2 is generated. We use the circuit to generate G (C), from which we generate 5 (C). 2.1 What is x(t)? Essentially, G (C) isthesumofall [-walks, which refer to walks of length[−1.(Each [-walk contains [ vertices, potentially repeated, and [ − 1 edges, potentially repeated. This follows standard graph theory terminology.) Each [-walk F is assigned angular frequency lF - thus each walk represents 8lFC 8lFC 4 . That is, G (C) = F 4 , where F is restricted to a possible [-walk according to graph = (+,). The eventual goal ofÍ this paper is to calculate the number of Hamiltonian paths. Therefore, it is essential to distinguish Hamiltonian paths from other [-walks. One simple implementation is to assign a unique frequency to walks sharing same vertices up to permutation. That is, in Figure 1, one possible 4-walk is A-B-A-C and another is C-A-B-A.
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