1. mst1tree模块

这个模块是输入距离矩阵c和pi值，给出mst1tree。
有两种方式：

from scipy.sparse import csr_matrix
from scipy.sparse.csgraph import minimum_spanning_tree 
# mst算法可以替换为自己实现

# 根据LKH的c代码实现
def getLT1(c,pi):
    c = (c.T+pi).T+pi
    sec_dist = np.hstack(np.sort(c)[:,2:3]) # 这里可以优化
    mst = minimum_spanning_tree(csr_matrix(c))
    r = mst.toarray()
    v = np.sum(abs(r)>0,axis=0)+np.sum(abs(r)>0,axis=1)-2
    return np.sum(r)-2*sum(pi)+max((v<0)*sec_dist)

# 这种方式是根据LKH report实现
def getLT2(c,pi):
    max_v = 0
    for k in range(len(c)):
        trans = np.hstack([k,np.arange(k),np.arange(k+1,len(distA))])
        nc = c[trans,:][:,trans]
        nc = (nc.T+pi).T+pi
        i,j = np.argpartition(nc[0][1:], 2)[:2] 
        mst = minimum_spanning_tree(csr_matrix(nc[1:,1:]))
        r = mst.toarray()
        v = np.sum(r)+nc[0][i+1]+nc[0][j+1]-2*np.sum(pi)
        if v>max_v:max_v = v
    return max_v
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其中minimum_spanning_tree模块，可以使用prim算法自己实现，cpu上性能和scipy差不多。

from numba import njit
@njit
def prim(matrix):
    n = len(matrix)
    non_matrix = np.zeros_like(matrix)
    v = np.zeros(n)
    v[0] = 1
    while sum(v) < n:
        mst_len = np.inf
        for i in np.arange(n):
            if v[i] > 0:
                for j in np.arange(i,n):
                    if v[j] == 0:
                        if matrix[i][j] < mst_len:
                            mst_len = matrix[i][j]
                            bi = i
                            bj = j
        v[bj] = 1
        non_matrix[bi][bj] = matrix[bi][bj]
    return non_matrix
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2. ascent模块

这个模块使用梯度下降法给出最后的pi值，首先给出个简单的版本：

def ascent_simple(c):
    pi = np.zeros(len(c))
    t = np.ones(len(c))
    best_w, v = getLT(c, pi)
    period = max(len(c)//2, 100)
    for k in range(period):
        w, v = getLT(c, pi+t*v)
        if sum(abs(v)) == 0:
            return w, pi+t*v
        elif w > best_w:
            best_w = w
            pi += t*v
        else:
            t /= 2
    return best_w, pi
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接下来是按照LKH_report的逻辑进行补全的版本，如下。这里去掉了initialPhase的判断。

def ascent(c):
    pi = np.zeros(len(c), int)
    t = 1
    best_w, v = getLT(c, pi)
    best_deg = np.sum(np.abs(v)) #np.sum(v*v)
    last_v = v
    period = max(len(c)//2, 100)
    while period > 0 and t > 0.5:
        period_continue = False
        for k in range(period):
            for i in range(len(v)):
                if v[i] == 0:
                    last_v[i] = 0
            pi = pi+t*(0.7*v+0.3*last_v)
            last_v = v
            w, v = getLT(c, pi)
            deg = np.sum(np.abs(v))#np.sum(v*v)
            if sum(abs(v)) == 0:
                logger.info("* T = %d, Period = %d, BestW = %.1f, Norm = %d" % (t, period, w, deg))
                return w, pi
            elif w > best_w or (w == best_w and deg < best_deg):
                logger.info("* T = %d, Period = %d, BestW = %.1f, Norm = %d" % (t, period, w, deg))
                best_w = w
                best_deg = deg
                t *= 2
                if k == period-1:
                    period_continue = True
            else:
                logger.info("  T = %d, Period = %d, BestW = %.1f, Norm = %d" % (t, period, w,deg))
                if k > period /2:
                    t = t*3/4
                    break
        if not period_continue:
            t = t/2
            period //= 2
    return best_w, pi
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3. generateCandidate模块

ascent模块给出来最佳的pi值，接下来用递推方式计算α-nearess值，以及生成candidate set。代码如下：

def geneCandidate(r, c, first_node):
    # 首先要将树结构用suc和dad两个数组保存下来，mst是用来遍历树的堆栈
    x, y = np.where(r > 0)
    alpha = c.copy()
    beta = np.zeros_like(c)
    for i in range(len(x)):
        beta[x[i]][y[i]] = beta[y[i]][x[i]] = c[y[i]][x[i]]
    neighbour = beta > 0

    suc = [0]
    dad = {0: None}
    mst = [0]
    heapq.heapify(mst)
    uv = set(range(len(c)))
    while len(mst) > 0:
        j = mst.pop()
        uv.remove(j)
        for nj in np.where(neighbour[j])[0]:
            if nj in uv:
                mst.append(nj)
                dad[nj] = j
                suc.append(nj)

    # 迭带所有的边
    for idx, i in enumerate(suc):
        if i == first_node:
            for jdx, j in enumerate(suc):
                if i == j or j == first_node:
                    continue
                if j == dad[i] or i == dad[j]:
                    alpha[i][j] = alpha[j][i] = 0
                else:
                    second_cost = np.sort(c[i])[2]
                    alpha[i][j] = alpha[j][i] = c[i][j] - second_cost
        else:
            for jdx, j in enumerate(suc):
                if idx >= jdx or j == first_node:
                    continue
                if j == dad[i] or i == dad[j]:
                    alpha[i][j] = alpha[j][i] = 0
                else:
                    beta[i][j] = beta[j][i] = max(beta[i][dad[j]], c[j][dad[j]])
                    alpha[i][j] = alpha[j][i] = c[i][j] - beta[i][j]
    candidate_index = np.argpartition(alpha, max_candicate)[:max_candicate]
    return alpha, candidate_index
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4. LinKernighan

这里仅放出简化版本，不使用Candidate，而是按顺序遍历所有点位。

def LKH(c, t):
    n = len(t)
    max_swaps = len(t) # 在没有优化的情况下，最多允许swap的次数
    for i in range(n): # 探索第一个点位
        nt = t.copy()
        x1 = (nt[i], nt[(i + 1) % n])
        G0 = c[x1[0]][x1[1]]
        for swaps in range(max_swaps):
            nt, G0, G, j = bestOptMove(c, i, G0, nt) # 探索第二个点位
            if G > 0.01:
                return nt, G
    return t, 0
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探索第二个点位的bestOptMove如下：

def bestOptMove(c, i, G0, t):
    n = len(t)
    best_g2 = -np.inf
    best_j = -1
    for j in np.arange(n): # 第二个点位也是按顺序遍历
        if (i + 1) % n != j and i != j and (i - 1) != j and (j + 1) % n != i:
            y1 = (t[(i + 1) % n], t[(j + 1) % n])
            G1 = G0 - c[y1[0]][y1[1]]
            if G1 > 0: # 探索条件：正收益原则
                x2 = (t[(j + 1) % n], t[j])
                y2 = (t[j], t[i])
                G2 = G1 + c[x2[0]][x2[1]]
                G = G2 - c[y2[0]][y2[1]]
                if G > 0.01: # 一旦找到优化的解，立即返回
                    t = TwoOptMove(i, j, t)
                    return t, G2, G, j
                if G2 > best_g2:
                    best_g2 = G2
                    best_j = j
    # 所有的j都不符合条件时，原样返回
    if best_j == -1: 
        return t, G0, 0, -1
        
    # 如果符合正收益原则的所有j都探索完都没有优化，那么选择其中最佳的j尝试进行swap
    return TwoOptMove(i, best_j, t), best_g2, 0, best_j
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接下来是数组版本的边交换代码：

def TwoOptMove(i, j, t):
    n = len(t)
    if i < j:
        t[i + 1:j + 1] = t[i + 1:j + 1][::-1]
    else:
        temp = np.zeros(n + j - i, np.int32)
        temp[:n - i - 1] = t[i + 1:]
        temp[n - i - 1:] = t[:j + 1]
        temp = temp[::-1]
        t[i + 1:] = temp[:n - i - 1]
        t[:j + 1] = temp[n - i - 1:]
    return t
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