Table 2.

Distances for the ITD problem with and without the use of our partition algorithm

OP	$|\mathrm{\Sigma }|$	With partition							Without partition
		Minimum distance			Average distance			Time	Minimum distance			Average distance			Time
		Min.	Avg.	Max.	Min.	Avg.	Max.		Min.	Avg.	Max.	Min.	Avg.	Max.
25	10	42	47.83	57	43.64	49.45	58.80	0.22	94	96.91	98	99.93	100.19	100.52	0.02
25	20	38	48.07	61	39.85	49.61	61.79	0.22	88	93.42	96	97.99	98.63	99.41	0.01
25	30	39	47.85	55	40.71	49.27	56.62	0.22	84	88.89	92	94.72	96.27	97.49	0.02
25	40	39	48.53	57	40.85	49.91	58.67	0.22	77	84.97	88	89.56	93.60	95.62	0.01
25	50	41	48.38	55	42.54	49.59	56.87	0.22	69	81.20	87	86.35	90.66	93.87	0.01
25	60	40	48.56	57	40.68	49.69	58.79	0.23	70	77.84	84	81.71	88.00	93.21	0.01
25	70	40	47.94	54	40.42	48.85	55.96	0.22	68	74.88	84	80.36	85.60	91.40	0.01
25	80	40	48.29	56	40.00	49.20	57.03	0.23	62	71.84	78	77.62	83.26	88.35	0.01
25	90	39	48.53	55	39.57	49.36	56.89	0.22	61	70.86	81	74.31	81.82	88.77	0.01
25	100	41	48.85	56	41.25	49.61	56.80	0.23	58	68.29	77	71.95	79.51	86.20	0.01
50	10	64	71.52	80	65.79	73.08	80.86	0.33	96	98.03	99	100.20	100.46	100.80	0.01
50	20	63	72.17	80	65.62	73.68	81.79	0.33	92	95.51	98	98.71	99.57	100.30	0.02
50	30	66	72.38	82	67.76	73.86	83.79	0.32	90	93.19	96	96.83	98.40	99.75	0.01
50	40	62	71.98	82	63.84	73.55	82.87	0.33	85	91.14	95	93.56	96.89	99.07	0.02
50	50	62	72.25	80	63.70	73.74	80.97	0.33	84	89.44	94	92.76	95.60	98.03	0.01
50	60	60	71.51	81	61.97	73.01	82.74	0.33	82	87.18	92	89.80	94.03	97.91	0.01
50	70	64	71.96	80	65.64	73.56	81.80	0.33	79	85.89	90	87.46	92.80	95.79	0.01
50	80	65	72.04	83	65.81	73.66	84.78	0.33	78	84.64	91	87.05	91.79	96.90	0.01
50	90	62	71.64	80	62.70	73.14	80.91	0.33	76	83.25	90	84.45	90.39	95.29	0.01
50	100	65	72.04	82	65.90	73.52	83.85	0.34	75	82.74	89	83.25	89.98	94.72	0.01
75	10	76	84.41	92	77.53	86.00	93.90	0.38	96	98.26	100	100.23	100.58	100.82	0.02
75	20	76	84.71	91	77.65	86.28	91.91	0.38	94	96.95	99	99.50	100.15	100.54	0.01
75	30	76	84.36	92	76.83	85.91	92.88	0.38	92	95.52	98	97.47	99.39	100.43	0.01
75	40	79	84.87	92	80.75	86.43	93.70	0.39	91	94.51	97	97.46	98.76	100.09	0.02
75	50	77	85.04	95	78.83	86.67	95.83	0.38	90	93.66	98	94.47	98.02	100.16	0.02
75	60	76	84.01	91	77.66	85.55	92.80	0.38	89	92.21	96	94.24	97.07	99.44	0.01
75	70	79	84.99	92	80.71	86.48	93.80	0.38	86	92.01	95	90.84	96.64	98.65	0.01
75	80	77	85.06	93	78.92	86.63	94.82	0.39	86	91.26	96	92.84	95.90	98.98	0.01
75	90	74	85.41	93	76.85	86.93	94.78	0.40	85	90.93	97	90.58	95.78	99.25	0.01
75	100	74	84.52	94	74.70	86.12	94.78	0.39	81	90.00	96	89.03	95.05	98.69	0.01
100	10	82	90.80	96	83.72	92.40	96.86	0.41	97	98.68	100	100.43	100.67	100.94	0.02
100	20	83	91.23	97	84.71	92.87	98.91	0.41	95	97.87	99	99.57	100.41	100.83	0.02
100	30	86	91.55	97	87.74	93.15	98.84	0.41	95	97.33	99	98.93	100.10	100.72	0.02
100	40	86	91.72	98	87.81	93.25	98.80	0.42	94	96.58	99	98.53	99.72	100.60	0.02
100	50	84	91.17	98	85.83	92.71	99.77	0.42	91	95.73	99	96.15	99.13	100.72	0.02
100	60	85	91.49	98	86.77	92.96	99.72	0.42	90	95.18	99	95.77	98.74	100.24	0.02
100	70	83	91.54	98	84.67	93.07	99.76	0.42	86	95.05	99	91.77	98.41	100.60	0.01
100	80	84	91.20	99	85.67	92.81	99.77	0.42	90	94.40	99	95.22	98.01	100.50	0.01
100	90	87	91.49	96	87.56	93.10	97.81	0.43	91	94.35	98	95.19	97.87	100.10	0.01
100	100	86	91.71	98	87.85	93.21	98.85	0.43	91	94.16	98	94.98	97.65	100.33	0.01